Exercise secrecy about source of terrible disease (10)
I believe the answer is:
pestilence
'disease' is the definition.
(pestilence is a kind of health problem)
'exercise secrecy about source of terrible' is the wordplay.
'exercise' becomes 'pe' (Physical Education**).
'secrecy' becomes 'silence' ('silence' can be a synonym of 'secrecy').
'about' is an insertion indicator.
'source of' indicates taking the first letters.
The first letter of 'terrible' is 't'.
'silence' placed around 't' is 'stilence'.
'pe'+'stilence'='PESTILENCE'
(Other definitions for pestilence that I've seen before include "dire horseman?" , "Deadly epidemic disease" , "evil force" , "Epidemic disease with a high death rate" , "Fatal epidemic disease" .)