Is it permissible to pray Asr about 10 minutes before its Adhan due to being late in returning home as Maghrib time approaches?

It is not permissible to perform the prayer before its prescribed time, except out of necessity for combining prayers during travel or rain. The questioner did not mention an excuse for performing it early, so it is not permissible for her to pray Asr until she is certain that its time has begun, even if the muezzin has not called the adhan (definition). A Muslim should be diligent in observing obligatory prayers and not prioritize worldly matters over them.