She asked the doctor to come check on her son again tomorrow.
Literal
Tomorrow once-more son [object-を] house-call-do [so-that-ように], she [topic-は] doctor [to-に] asked.
往診する ('to make a house call') is a medical compound — 往 (go to) + 診 (examine). ~てくれるように頼む means 'to ask someone to do X (as a favor),' layering ~てくれる (benefactive) with ~ように (purpose) and 頼む (to ask/request).