Is it permissible to use the mobile phone for playing after finishing studying, without the mother's knowledge, if it was requested from her for studying purposes?

It is permissible to use the mother's mobile phone for permissible play after finishing studies, if the mother explicitly or customarily permits it. However, one should utilize time for what benefits them in this world and the hereafter, and avoid wasting it in amusement and play, for a Muslim is accountable for their time.