My question is the following:

Can a (probabilistic, deterministic, ndtm) oracle turing machine $A$ calling an oracle residing in a superior (more difficult) complexity class $B$, have less power then the oracle called ($A^B < B$).

My believe is that this new class should be at least as powerful as $B$?

For example, assuming that i've $P^X$, where $X$ resides in a superior complexity class.

Could $P^X$ be weaker than $X$ in some cases and, if yes, is a meaningful complexity class?