I would say that his statement is probably a bit overreaching. I've not read Fenby's book so I'm unsure of the exact context in which he's making that statement but your assumption from it was probably correct among a few units at least. Let's break it down a bit and get an image of the CCP Red Army following the 1927 split of the United Front.

The Early Years, 1923-1927

Soviet influence and intervention on Chinese affairs after the Bolshevik revolution was quite immediate. Even as the Russian Civil War raged on in the early 1920s, communist politicians and military advisors urged action in China, especially as the KMT made progress in Guangdong. There was a lot of interest in the KMT about learning from the Soviets, who had established the first successful "revolutionary army" in history. On August 26, 1923, Sun Yat-sen had dispatched a delegation to the USSR (named the "Doctor Sun Yatsen Delegation") to inquire about Soviet organization, training and equipment. They visited military academies and discussed the planning of establishing a military academy in southern China, which would become the Whampoa Military Academy, where many KMT and early CCP leaders were trained (Whampoa still operates today as a military academy). Under the leadership of Mikhail Borodin, these early years saw not just the martial development, but also the beginning of a complex political system that engaged in revolutionary organization and theory. It was here that the officers of the Peasants Red Army (which later formed the famous Eighth Route Army) received their training for years until the 1927 schism.

While Whampoa was a popular destination for many future famous leaders in China, some officers who joined the CCP through the KMT already had tons of military experience. Zhu De, for example, graduated from the Yunnan Military Academy and fought under Cai E during the Anti-Monarchist War in 1915. Other generals, such as Peng Dehuai, served in various warlord armies. So many of these officers who would come to make up the backbone of the Red Army were fairly seasoned. In fact, at this point in time, they would've been some of the most experienced with actual combat. While the West was trained by WWI, these Chinese military leaders were fighting near constant domestic conflicts from 1915 up to the end of the Civil War in 1949. The Soviet influence introduced more formal and modern techniques, but by the 1920s many of the old guard leaders were experienced in combat.

The Northern Expedition was executed with great success by these leaders. Many communist units partook in the unification, but we should make a quick note that the structure of the KMT at this point would've been something along the lines of this: the military wing was dominated by the rightist bloc of the KMT, with something like 75% of units being led by what we'd call nationalists, and around 25% communists. So when the schism came in 1927 Shanghai, it didn't just take the CCP by surprise, it also meant they were outnumbered by a lot. Caught in the chaos, Soviet advisors were expelled from the party and the CCP units were in some instances entirely decimated.

Jiangxi Soviet, 1931-1934

In 1931, Mao and a few of his loyalists formed the Jiangxi Soviet, where they fled to Ruijin in the aftermath of the United Front betrayal. While the political workings of the Soviet are interesting on their own, the war that would wage these three years and beyond are incredibly complex. Some Soviet advisors did follow Mao into Jiangxi. And I think this may be the most relevant part of your question; now that the CCP needed fresh recruits after nearly being vanquished, the Red Army had a lot of units with little to no combat experienced enlistees. In addition to this, many of these poor peasants that formed the backbone of the new Red Army had no experience in the military either. Things were highly irregular. Its entirely possible that some of these units were taught very basic maneuvers by Soviet advisors. But far more likely most were taught by their own Chinese leaders. The Red Army still retained many great experienced generals, it just needed men to fill the gap. But war in Jiangxi became increasingly brutal and cruel. It became far more guerrilla too. Keep in mind that Mao himself was a military leader for most of the early part of his life. Early CCP doctrine kept in line with Soviet and KMT doctrine; i.e., conventional warfare. But now, Mao realized convention would not work against the KMT, they held the upper hand in that respect. Jiangxi is mountainous and was very rural, it was an ideal location to fight until a better opportunity came (it was also a good location to organize and radicalize poor peasants who made up the majority of the population). The ecstatic volunteers made up mostly from the poor peasantry soon found themselves thrown into battle; throughout those three years, Chiang Kai-shek would lead five "encirclement campaigns," and hundreds of thousands were most likely killed. It was a brutal war that would only be matched in severity by the Japanese invasion just three years later. The guerrilla war saw Red Army units engaging both KMT armies as well as *baojia-*style militias funded by rich peasants and landlords who allied with the KMT.

The general CCP strategy was to lure KMT units into areas of Jiangxi that were under their total control, where they would "strengthen their defenses and clear the fields," evacuating civilians and burning property. Some KMT soldier POWs had recalled going days without eating, embarking on forced marches for days, and then getting ambushed by Red Army troops. Early in the war, the Red Army was split into full-time regular armed forces, with part-time local militias. Many of the militias took an auxiliary role, helping with spying, medical care, logistics, etc. The Red Army soon had itself an experienced core of guerrilla troops. According to Marc Opper:

KMT forces would set up camp in a village for the evening. When night fell, CCP forces would open fire with large, loud cannons on the KMT's positions. KMT forces directed machine-gun fire toward what they thought were CCP positions, but would remain firmly within the village. In the morning the CCP forces would retreat to a nearby hill or mountain as the KMT sent a few small units out in search of CCP forces. When KMT forces were marching they were often the targets of far-off sniper fire... As one CCP veteran recalled many years later, when the KMT entered areas under CCP control "they found no food to eat, they could not get any rest, they could not gather any intelligence, and they could not find guides. They were drowning in the ocean of our people's war."

Combined with the practice of evacuation civilians the CCP deemed "unreliable" into the heart of the Soviet away from the KMT, was effective and the CCP enjoyed control over the population until 1933. During 1933, Mao lost power within the CCP and was replaced in a military capacity by Zhang Wentian, Bo Gu and a German advisor named Otto Braun. They concluded that the Soviet had reached the point where they could successfully switch from guerrilla to conventional warfare. In a WWI-style manner, the KMT and CCP both set up trenches and blockhouses (pillboxes) along with other fortifications facing each other. The issue was that the KMT had artillery and air-support, while the CCP did not. The troops were often sitting ducks when fired upon. Red Army units moved to attack KMT defenses in urban cities and villages where they had been concentrated, resulting in catastrophic loss of life. By the end of 1934, most of the Red Army units departed on the Long March. Jiangxi then became an ideological battleground until 1937.

Conclusion

I rambled a bit, but all this was to say that the Red Army, from early on, was a very experience and trained army. They didn't really need advisor training after 1927, what they really needed was materiel aid. Fenby's statement may be true, but it was most likely true at those points when the Red Army was recruiting from new peasantry, and by the 1930s this training would have been mostly conducted by Chinese officers, not Soviet ones.

Sources

The Establishment of Chinese Military Academies & the Soviet Aids in 1920s-1950's, Zhao Yanghui

People's Wars in China, Malaya, and Vietnam, Marc Opper