Using the context of the sentence (the person says they hate cleaning their room, then use 'but' to interject which is used for contradiction/juxtaposition) and comparing to other words with the same prefix of "ac" (accept, acquire), I can conclude acquiesce means to "reluctantly accept".
“Acquiesce” means to reluctantly agree to do something. The evidence I used to come to that conclusion is that the person first says they hate cleaning their room! However, when their mothers asks them to do so, they agree to do it anyway. It’s pretty obvious that this isn’t going to be something they want to do. Therefore, “acquiesce” means to agree reluctantly.