Dicky, Sam, Lee and Sierra get illness (7)
I believe the answer is:
measles
'illness' is the definition.
(measles is a kind of health problem)
'dicky sam lee and sierra' is the wordplay.
'dicky' is an anagram indicator (dicky can mean not functioning properly).
'and' says to put letters next to each other.
'sierra' becomes 'S' (phonetic alphabet: alpha, bravo, charlie etc.).
'sam'+'lee'='samlee'
'samlee' is an anagram of 'measle'.
'measle'+'s'='MEASLES'
'get' is the link.
(Other definitions for measles that I've seen before include "Children's viral disease" , "Morbilli is technical term for this illness" , "Childhood malady" , "Infectious disease covering body with red spots" , "Rash-causing disease" .)