No. Power of attorney does not give you access to an inmate's phone and messaging records, and neither does paying the bill.
Inmates retain privacy rights over their communications, at least from other civilians. The facility monitors calls and messages as a matter of policy, and that is disclosed to everyone using the system. But that monitoring belongs to the institution, not to you. Being the person funding the account does not change that.
The harder truth here is that if the real question is whether he can be trusted, no phone record is going to settle that for you. People who want to hide something find ways to do it. And if the concern has reached the point where you are looking for ways to surveil his activity, that is worth sitting with honestly.
If you do not trust how he is using what you are paying for, the most straightforward move is to stop paying for it. That is not a punishment; it is a boundary. You are not obligated to fund communication you have doubts about, and protecting your own peace of mind matters.