Take another look at the sentence:
Hoping to relax after a busy week at work and school, Cheryl smoothed on some sunscreen, settled onto her pool raft, and opened her new Stephen King novel, then her neighbor cranked up the lawnmower, ruining the quiet.
Hoping to relax after a busy week at work and school is a present participle phrase. Cheryl smoothed on some sunscreen is a main clause. You need a comma when you attach a participle phrase to the front of a main clause.