Thanks in advance.
Personally, I don't know if air superiority was ever dismissible in any fight. However, it was far from the dominant factor in all campaigns. Of course WWII showed that airpower could smash apart enemy supply lines. It could lend a massive hand of help to an armored spearhead or defensive force. It was also useful in reconnaissance and factory bombing. But there were many limitations on what a Messerschmidt, Zero, Spitfire, IL-2 or Mustang could do.
So the absence of omission of air superiority is probably not quite what you're looking for, but it was interesting to note that WWII is probably one of the last global wars to be fought in which air power would be entirely absent from some battles. Planes were still relatively fragile back in that era, and they could not operate well in cold or rain conditions.
So large sections of the Battle of Moscow, Stalingrad, the Ardennes Bulge and the Finnish wars were fought without air power. Authors like Stahl, Glantz and Beevor mention the major occasions when airpower could not be used under -0 temperatures. The early Soviet counteroffensives depended on using weather to negate the Germans' airpower, and put the battle back into the constraints of artillery, tank and infantry warfare. Moreover, much of their success in operations like Bagration and Romania depended on the Soviets masking their troop movements and preventing Luftwaffe responses. Would airpower be considered dominant then, if it is foiled from its most useful skill: reconnaissance, by thorough strategic planning?
Almost all urban fighting from Monte Cassino to Stalingrad to Berlin could not be dominated by air power either. Dive bombing was too comparatively weak, heavy bombing too crude and inaccurate to be of much use in urban warfare. Bombs could ruin a city, but turn it into a dangerous warren of rubble. Places like Pavlov's House in Stalingrad or the Reichstag in Berlin, simple buildings which would be busted into dust by any drone nowadays, resisted battalions for weeks on end.
It's also important to remember that Europe was relatively clear and easy to use planes within, and that America's war on the Pacific islands was in many senses uniquely suitable for air power dominance. In a different location, such as a forested jungle, the air power that was available in the 1940s was incredibly limited. As 'Touched With Fire' mentioned, the earliest phase of the war in the Pacific was fought in a limited but deeply forested region of Guadalcanal and New Guinea. The Japanese lost because seaborne logistics were cut apart by planes and u-boats, but the battles themselves tactically went to whichever side threw the most men in with the best weaponry. In a different location less dependent on the sea like Burma or Vietnam, air power was not of as much use, as the early Japanese advances showed.
To conclude, WWII was one of the last total wars where air power was severely constrained by major factors of terrain, weather and accuracy. Because of this, there were many important campaigns where aircraft were used minimally.
I'm not sure if this is exactly what you're looking for, but the naval battles during Guadalcanal were very interesting in regards to air power. For the total story, James Hornfischer's Neptune's Inferno covers the campaign.
The basic story is this. The Americans controlled the entire island and surrounding sea during daylight. The Japanese controlled the area at night. There were a number of factors that led to this. First, the absence of aircraft carriers in the warzone. The Japanese had just lost 4 fleet carriers and could only muster between 1 and 2 carriers at any given time. The Americans were down to a single carrier (the Hornet) in the fall. This made the only land base, at Henderson Field, even more proportionally dominant. The Japanese were afraid to lose their carriers to land-based air power, and the Americans figured they could not lose a carrier by using Henderson Field.
Second was the differential abilities of the US and Japanese and night fighting. The Japanese had much more night fighting training and their discipline made coordinating nighttime operations more smooth. They dominated the night battles but were unable to make the breakthru they needed to gain naval superiority. They could not resupply or bombard the Americans to the level needed for victory. As the Americans learned how to use ship-based radar, the nighttime battles turned in their favor. In addition, the ability of the Americans to conduct naval operations unimpeded during the day was a major factor in them wearing down the Japanese Army and Navy.
The Battle of Hürtgen Forest is a prime example. I alluded to this in a previous answer
The vast superiority in air support that the United States had was effectively neutralized by the rugged terrain and very poor weather. The Germans made no use of air support in the battle. The fall of 1944 was notoriously wet and gloomy, with most days in October and November being quite cold and rainy, with solid overcast; the first snow fell even before Thanksgiving. In this area along the German-Belgian border, the terrain is mostly hilly with a few flat areas, cut by steep valleys through which flow creeks or rivers. The banks of the valleys and some open areas beside them are heavily forested. The towns and villages in this area are mostly located on the bald tops of these hills, and mostly produce root crops like carrots or potatoes. German pillboxes were scattered throughout the forest, some of them located next to villages. Prepared ones were mostly made of concrete, although some were constructed of large logs.
The trees in the forest were planted closely together in neat rows, almost like soldiers on parade. The lower limbs of the trees interlocked at a low height, so that soldiers often had to stoop in order to walk. Effective control of units larger than platoon size was impossible. Fighter-bombers found it very hard to hit targets in the thick forest. Combined with the poor weather, air support was ineffective. P-47s and P-38s often had to orbit above the overcast until holes in the clouds opened up for them to dive through.
Just because the American advantage in air support was greatly curtailed doesn't mean it wasn't there at all. The P-47s of the 365th Fighter Group were able to fly 45 missions in October; the 404th Fighter Group was able to fly 20 out of 30 days in November. The 370th Fighter Group earned a Distinguished Unit Citation after napalming the town of Bergstein in bad weather and under heavy antiaircraft fire on December 2nd.
"The next day the Group ran an uneventful sweep over the Ninth Army front in the morning. The weather was poor, with a solid low overcast at 5,000 feet, and high cirrus around 20,000, but not enough clouds at the right altitudes to encourage the Luftwaffe."
"November 9th the 506th and 507th worked over a bridge across the Roer at Linnich, funnel for two main highways. The 508th off last, ran into low ceilings, rain, sleet and snow, but managed to unload its bombs on a marshalling yard at Bergheim, half way between Julich and Cologne."
"As the Group returned to base, however, a hazard more dangerous than flak arose. The clouds had settled-down to 800 feet over A-92, and landing aircraft were peeling off, disappearing into the "soup" and trusting to luck there wouldn't be another plane in the way when they broke out under the 800-foot layer. There were wild, crazy landing patterns, elements cutting in front of aircraft on the final approach, 48th Group planes mixed up with the 404th squadrons coming in scattered."
Sources:
The Guns at Last Light, by Rick Atkinson
Hell Hawks!: The Untold Story of the American Fliers Who Savaged Hitler's Wehrmacht, by Robert F. Dorr
365th Fighter Group
http://www.368thfightergroup.com/368-timeline-w.html
370th Fighter Group
http://www.armyaircorps.us/370th_Fighter_Group.cfm
404th Fighter Group History
The Gothic Line was a major campaign during the Italian campaign, and one of the main and most brutal battles of this campaign was the Battle of Monticelli, which was fought between September 14 - 18, 1944. The Germans had built a network of fortifications deep into the mountain of Monticelli, making it impossible to attack it from the air.
The mountain was taken inch by inch by the courageous men of the 363rd Infantry Regiment, while nearby Mount Altuzzo was being assaulted at the same time by the 362nd Infantry. It was brutal bunker to bunker fighting (often resulting in hand-to-hand combat) on a rather steep mountainside with very little cover. Casualties were extremely high on both sides. F Company of the 363rd Infantry Regiment suffered a 37% casualty rate, in comparison Omaha beach had a casualty rate of 41% on D-Day.
Excerpts from pages 70 - 83 of 'History of the 363rd Infantry Regiment: One Regiment Of The 91st Division In World War II' by Ralph E. Strootman:
"F Company crossed the ravine early the next morning, September 15th, 1944, and prepared to continue the attack up Monticelli. After completing the crossing of the minefield that morning F Company had attacked up the slope with Lieutenant Hatcher’s 3rd Platoon on the right, First Sergeant McKelvey’s 2nd Platoon on the left, meeting heavy resistance in the form of an estimated eight enemy pillboxes across their front.
Machine-gun fire from these emplacements caused serious casualties, especially in the right platoon which was cut down to nine men and a sergeant. Lieutenant Hatcher was one of those who lost his life. Captain Crowden committed his reserve 1st Platoon to the right to relieve the handful of the 3rd Platoon which was then withdrawn 100 yards to the rear of the 1st and 2nd Platoons. The company mortars had been emplaced and the two light machine guns attached to the 2nd Platoon; now the company got down to the dirty business of systematically knocking out the emplacements which faced them.
The next day, September 16th, F Company began to receive heavy fire from the left front, which held up the attack. The company could not advance more than 300 yards from the line of departure during daylight, and the attack was held up until nightfall. As nightfall came, the company continued its dirty business of knocking out the enemy emplacements on Monticelli.
On September 17th, Sergeant McKelvey began inching his platoon forward by fire and movement. One of his squad leaders went down and the assistant was killed almost immediately as he went to take command. Sergeant McKelvey wormed over to the squad, reorganizing it. “We just couldn’t move,” explained Private Glen H. Wells, “because we were getting so much fire from the pillboxes. Sergeant McKelvey told us to hold up and he began moving forward himself – the tracers were cutting right over him. Then he had to crawl through the wire.” One bullet tore two holes in McKelvey’s helmet as he worked his way through the barbed wire and disarmed an S mine while the squad kept trying to button up the enemy emplacement.
“Private James R. Wixon was firing at the machine gun but he was killed,” continued Private First Class Theodore J. Drozdowsi. “Sergeant McKelvey kept moving and when he got close enough he threw a Molotov Cocktail, then ran and jumped into the pillbox. I heard him hollering at the Germans and the next thing I knew five Jerries came running out of the hole with Sergeant McKelvey behind them. He sent them back and we never had any more trouble from the pillbox after that.” Sergeant McKelvey was later commissioned as a second lieutenant. He was later awarded the Distinguished Service Cross for this action.
While the mountain of Monticelli was captured at this point by members of K Company, F Company was still faced with the network of fortifications that they had been reducing for the past several days. Eventually as the fight progressed throughout the night the remaining German elements withdrew, and Monticelli was officially secured. On the morning of September 18th F Company, along with the 363rd Infantry remained dug in on Monticelli and continued the evacuation of their wounded."