Tweaking textbook RSA to make the encryption a Pseudorandom function
This is not a secure pseudorandom function.  No tweak is going to make this a pseudorandom function.  After all, it is a public-key operation, so anyone who knows your algorithm can compute $f(m)$ given $m$.  Consequently, it cannot possibly be pseudorandom: there will always be a trivial distinguisher.

What are you actually trying to accomplish?  Why do you think you want a tweak to RSA that makes it a pseudorandom function -- what problem are you trying to solve?  I think you may need to step back from the specific mechanism you have in mind and think through what are the broader requirements of your particular application.  Once you've done that, we may be in a better position to help you solve your problem.