Foolishly Dominican's beginning to ignore suffering from sleeplessness (9)
I believe the answer is:
insomniac
'suffering from sleeplessness' is the definition.
'foolishly dominican's beginning to ignore' is the wordplay.
'foolishly' is an anagram indicator.
'beginning to ignore' means to remove the first letter.
'dominicans' with its first letter taken off is 'ominicans'.
'ominicans' with letters rearranged gives 'INSOMNIAC'.
(Other definitions for insomniac that I've seen before include "Habitually sleepless person" , "One who can't sleep" , "Person finding it difficult to sleep" , "who won't get off?" , "One unable to sleep" .)