I wonder, does PLA has Military Police units? Or the PAP is also Military Police for PLA personell?
PAP is Gendarmerie without organizational jurisdiction or responsibility within PLA. In a way you can say PAP is just another branch of PLA which assist civilian government.
There is no
formal internal organization that carries out Military Policing function. A policing unit within Chinese military is
against Communist ideology because the military is meant to be people's army driven by ideology commitment and guided by political indoctrination (the political commisars and CCP party cells). Having a police unit would defeat that funding principle in that the soldiers are not self motivated and committed but rather contractors who need some sort of law inforcement.
This is the principle.
In practice,
Outside of barracks there can be an
ad-hoc group of soldiers who is charged to carry out disciplinary duties. They are called "纠察"/"Jiu Cha". You may see them on the street to watch out soldiers dress and behavior outside of barracks in big cities during holidays, or regularly in Beijing. In this case, both PAP and PLA have "Jiu Cha" troopers and they have jurisdiction over one another as both PAP and PLA personnel are equally soldiers.
Here is the "Jiu Cha" drawn from PAP members. See the helmet badge is of PAP.
Here is the "Jiu Cha" drawn from PLA members in Tiananmen square. See the helmet badge is PLA.
You can also see the two Chinese characters "警备" on the helmets of both PLA and PAP. The words designate troops that carries out local defense of major cities. They are mostly part of PAP except in Beijing where they are part of PLA. Either way, they carry out the same duty.
This again tells that PAP and PLA is very closely related.
Within barracks, "Jiu Cha" does not exist. Any soldiers may and will carry out disciplinary duties put on them at any time by their commanders. I was once charged to be a "Jiu Cha" trainee during my military training to experience how life is in the barrack, that role rotates among persons who is well regarded as good soldiers (disciplined, well dressed, well walked etc.).