Here is the fragment again:
Kissing the wet frog in hopes of getting a cool prize.
You correctly identified the fragment as a participle phrase. Kissing, the present participle that begins the phrase, is the clue.
To fix this fragment, you could add a subject and verb, modifying the phrase like this:
Despite the jeers from the crowd, Petra kissed
the wet frog in hopes of getting a cool prize.
Or you could use Punctuation Rule 3:
Main Clause + , + Participle Phrase.
The correction would look like this:
Petra said a prayer and puckered her lips,
kissing the wet frog in hopes of getting a cool prize.
Or you could use Punctuation Rule 4:
Participle Phrase + , + Main Clause.
And the correction would look like this:
Kissing the wet frog in hopes of getting a cool prize,
Petra gagged and wiped her mouth with her shirt
sleeve.
Wear that crown with pride!