Given the choice, the ending "-ly" tells you that it can only be an adjective or an adverb. While there are a few adjectives ending in "-ly" (e.g. friendly, costly, monthly, silly, holy, etc.), most of them are adverbs.
The way how you can find out whether it is an adjective or an adverb is by placing the word into these two sentences to find out where it fits:
(1) This is a/ an _____ person/ thing. => if it fits, it's an adjective, because it describes a person or thing.
(2) This can be done _________. => if it fits, it's an adverb, because it describes a verb
OR: This is __________ nice/ bad/ interesting. => if it fits, it's also an adverb, because it describes an adjective.
Now try this with "entreatingly":
(1) This is an entreatingly person/ thing. => WRONG => this is why it's not an adjective
(2) This can be done entreatingly. => CORRECT
OR: This is entreatingly bad. => CORRECT
=> Therefore, it is an adverb.
Just remember, adjectives describe nouns (people or things), i.e. how they are, whereas adverbs describe verbs (how you do something) and adjectives.
