The full sentence is 「寄る辺ない侘しさに」. I did some research and found out that「に」is used with na-adjectives to basically turn them into adverbs, but「侘しい」is an i-adjective, so it doesn't make much sense.
First of all, is this really a full sentence? I googled and found this song. If your phrase is from this, 寄る辺ない侘しさに is a part of a sentence, not a full sentence. Please always provide the entire context.
This に is a cause/reason marker that is used with a word related to emotion or psychological reactions.
- に in コントロールに苦しまなくなった
- When do you use に to indicate a cause or reason for something?
- Grammar of nounの多さに
- Meaning of に in この狭さに
侘しい is an i-adjective, but note that it has already been nominalized with さ ("-ness") here. 侘しさ is a noun meaning "loneliness" or "melancholy".
(literally) If it's like you rest your cheek in your hand because of desolation with nowhere to turn, things won't change.
→ When you feel desolate with nowhere to turn, just resting your cheek in your hand won't change anything.
Even if your 寄る辺ない侘しさに is not from this song, に is still a cause/reason marker. If this is a book title, the implicit focus is how someone psychologically reacts to the 侘しさ, so it might be translated like "Loneliness Makes Me..." or "In Desolate Solitude...".
As noted in the comments, に is operating on a noun, in this case it means Because of / From. The sentence is incomplete.