Heal Charlie badly with my dubious branch of science (7)
I believe the answer is:
alchemy
'dubious branch of science' is the definition.
(supposed science of the Middle Ages)
'heal charlie badly with my' is the wordplay.
'charlie' becomes 'c' (phonetic alphabet: alpha, bravo, charlie etc.).
'badly with' indicates an anagram.
'heal'+'c'='healc'
'healc' with its letters mixed up with 'my' can make 'ALCHEMY'.
(Other definitions for alchemy that I've seen before include "Pre-scientific chemistry" , "Philosopher's Stone its object" , "Chemistry's forerunner" , "Medieval chemistry to turn base metal to gold" , "Forerunner of eleven ac" .)