Is there any way to request my employer to pay 32% taxes from that month until the end of the year,
according to what I just read on the subject you should let your employer know you want them to deduct higher percentage tax (32) - but the article didn't state if it should be made as a formal note/letter. My wild guess is that telling them and making yourself clear/insisting about it is enough.
if your second job is a free-lance (działalność gospodarcza) you don't have to pay additional ZUS as far as I recall - but if it is a regular employee position further ZUS will be deducted percentage wise