I'd like to ask you to look after my dog while I'm away.
Literal
Absence [possessive-の] during, my dog [possessive-の] care [object-を] request want [explanatory-んですけど].
~をお願いしたいんですけど is a layered polite request: お願いしたい (want to request) + んです (explanatory background) + けど (trailing 'but' for softening). Very natural way to ask a favor.