Learn English – Is “schwa” sound voiced or unvoiced sound

"Put on" is the verb phrase, and "them" is the object. So "put them on" is not a phrasal verb in itself.

"Put it on", "put that on", "put this on", etc. are all valid and common expressions.

I think the phenomenon you are describing is not limited to the verb phrase, but to the sentence as whole - many short and not-so-important words will have sounds cut off, merged, or not pronounced completely/"schwaed". Since "on" significantly changes the meaning of "put" though (when put is used reflexively), it's important and often stressed or said correctly. "On" would be said less clear in most spoken sentences if the meaning was "put it on the table", for example.

FWIW Forms of be and have are vulnerable to this especially since they are often contracted by virtue of often being auxillary verbs.