Would PPP adjustments between countries be irrelevant if all countries had the same relative prices? I'm trying to understand the relationship, if any, between the fact that countries have different relative prices (e.g. different 'food price-to-general consumer prices' ratios) impacts the need for purchasing power parity adjustments between countries.
That's correct, the PPP adjustments are made to account for the fact that living in some places is more expensive than in others. So if you are comparing two places with equal relative prices, their PPP index should be the same, and so it is irrelevant if you use the adjustment or not. 

Note that saying that prices are the same in different countries can be quite tricky, since typically goods that can be found in some countries are not even sold in other countries. But in principle, the PPP is trying to adjust for the fact that the same 100 American dollars have a larger purchasing power in some countries than in others.