Periodically, varsity hymn gets a complaint (6)
I believe the answer is:
asthma
'complaint' is the definition.
(asthma is a kind of disease)
'periodically varsity hymn gets a' is the wordplay.
'periodically' indicates alternate letters (letters must be selected 'every so often').
'gets' means one lot of letters go next to another.
'varsity'+'hymn'='varsityhymn'
The alternating letters of 'varsityhymn' are 'asthm'.
'asthm'+'a'='ASTHMA'
(Other definitions for asthma that I've seen before include "Respiratory problem" , "Shortage of breath" , "Cause of breathing difficulty" , "breathing problems" , "inspiring trouble" .)