The role of the protecting power was delineated by the Geneva Convention Relative to the Treatment of Prisoners of War (1929), which was the standard of international law for the treatment of prisoners of war at the time, and to which all of the major combatants of WWII (aside from the Soviet Union) were signatories. Article 86 of the Convention states that:

"The representatives of the protecting Power or their recognized delegates shall be authorized to proceed to any place, without exception, where prisoners of war are interned. They shall have access to all premises occupied by prisoners and may hold conversation with prisoners, as a general rule without witnesses, either personally or through the intermediary of interpreters. Belligerents shall facilitate as much as possible the task of the representatives or recognized delegates of the protecting Power. The military authorities shall be informed of their visits."

On paper, the protecting power had a pretty broad remit to inspect prisoner of war camps, but there was no independent authority that could enforce this mechanism if a belligerent or belligerents violated this article. That said, in general, the belligerents did respect the requirements of the Geneva Convention and allowed visitation by the representatives of the protecting powers (Switzerland or Sweden in most cases). Of course, the notable exception was German camps for Soviet POWs, who were treated terribly, with complete disregard for the norms of international law; 58% of Soviet prisoners in German captivity died. The Germans used the excuse that the Soviet Union was not a signatory of the 1929 Convention, but that argument was untenable, since the Convention required signatories to treat all prisoners of war according to the terms of the Convention whether their co-belligerents were signatories or not. It should be noted, however, that the Soviet Union also rejected Red Cross efforts to mediate with Germany regarding their prisoners of war.

In the other cases though, the protecting powers carried out a wide variety of supervisory and support activities, primarily through the Red Cross, but also via other international organizations like the YMCA. The delegates visited at relatively regular intervals and spoke with the appointed leaders of each national group (known as men of confidence, from the French term homme de confiance). They took notes on the men of confidence's reports about conditions in the camp, including their requests for various types of supplies. The inspections became less regular toward the end of the war, when many of the prisoner of war camps in Germany were in or near active combat zones. However, we know from postwar statements of repatriated prisoners that conditions deteriorated across the board during this period.

Unfortunately, the Red Cross inspectors had no real authority to actually enforce the Geneva Convention, so all they could really do was register the complaints of the prisoners and urge the detaining power to address them. Although the belligerents generally tried to maintain good conditions for prisoners of war in hopes of ensuring reciprocal treatment for their prisoners, there was no enforcement mechanism to punish violations.

One of the most important functions the Red Cross performed was supplying food parcels to the prisoners every few weeks, which included things like tinned meat and chocolate that the prisoners used to supplement their generally rather bland rations. These parcels became even more important late in the war, since supply lines and logistics were disrupted and food supplies in the camps became inconsistent. Allied prisoners also reported that they shared food from these parcels with the Soviet prisoners in camps that held both Western Allied and Soviet prisoners; the Soviet prisoners didn't receive food parcels, and they always received the worst food rations in the camps, so whatever the Allied prisoners could share made a significant difference to them (although sharing food like this was formally forbidden by the German camp authorities).

The Red Cross and YMCA also provided supplies to the prisoners to allow them to carry out recreational and cultural activities. These included books for the camp libraries, educational materials, musical instruments, and sporting equipment, which allowed for the maintenance of regular recreational activities in the camps. If you read the reports from Red Cross and YMCA delegates, they often include requests from prisoners for these types of supplies.

So, basically, the protecting power had the right to inspect the camps and was able to carry out charitable activities in support of the prisoners, but had no real authority to enforce the provisions of the Geneva Convention and couldn't do much beyond registering prisoners' complaints when those provisions were violated. They also had no authority to prevent the worst abuses of prisoners of war, which were primarily targeted toward Soviet prisoners of war in Germany. I know this answer is very Eurocentric, and I'm sorry for that, but that's my area of knowledge and I can't really speak to what happened with prisoners of war in the Pacific Theater (although the treatment of prisoners of war by the Japanese obviously did not comply with the standards of the Geneva Convention).

Primary sources:

Convention Relative to the Treatment of Prisoners of War, Geneva, 27 July 1929, Article 86-87 (text can be found here)

National Archives and Records Administration, College Park, MD (RG 59, 153, 289, 389) [Red Cross camp reports; also held at ICRC Archives in Geneva and the British National Archives]

United States Holocaust Memorial Museum Archives, Washington, DC (RG-30.007M) [YMCA camp reports]

Secondary sources:

Howard Levie, "Prisoners of War and the Protecting Power," American Journal of International Law 55, no. 2 (1961).

Andrei Şiperco, Crucea Roşie Internaţională şi România în perioada celui de-al Doilea Război Mondial: 1 septembrie 1939-23 august 1944: prizonierii de război anglo-americani şi sovietici, deportaţii evrei din Transnistria şi emigrarea evreilor în atenţia Crucii Roşii Internaţionale (Bucharest: Editura Enciclopedică, 1997)

The United States Holocaust Memorial Museum Encyclopedia of Camps and Ghettos 1933-1945, vol. IV: Camps and Detention Sites under the German Armed Forces (Indiana UP, 2022)