Letter from desperate elites clinging to power (7)
I believe the answer is:
epistle
'letter' is the definition.
(epistle is a kind of letter)
'desperate elites clinging to power' is the wordplay.
'desperate' indicates anagramming the letters (I've seen this in other clues).
'clinging to' indicates putting letters inside.
'power' becomes 'p'.
'elites' anagrammed gives 'eistle'.
'eistle' placed around 'p' is 'EPISTLE'.
'from' acts as a link.
(Other definitions for epistle that I've seen before include "(E.g. Paul's) letter" , "may be written off" , "Paul's work" , "a for instance" , "Apostle's letter" .)