Yes!  It depends on the retention mechanism (mobile phase, column...).

Don't forget in the C=C there are electrons available for complexing/bonding/retention

Yes, given your limited details the polymer-drug will have more chances for the hydrophobic interaction. 

So, can the higher Mw of the conjugate be the reason why it comes later even it has a hydrophilic character?

As a general rule, hydrophilic molecules are expected to come faster than hydrophobic ones on Reversed Phase Adsorbtion Chromatograpy, aren't they?

You said the polymer was amphiphilic. So yes it will contribute towards retention