What Kate went through, thanks to Mike in Gateshead? (6)
I believe the answer is:
taming
'what kate went through' is the definition.
(Kiss Me, Kate is based on The Taming of the Shrew)
'thanks to mike in gateshead?' is the wordplay.
'thanks' becomes 'ta' (UK informal version of 'thank you').
'to mike' becomes 'm' (phonetic alphabet: alpha, bravo, charlie etc.).
'gateshead?' becomes 'g' (head letter of 'gates').
'ta'+'m'+'in'+'g'='TAMING'
(Other definitions for taming that I've seen before include "for domestication" , "Correction by discipline" , "Overcoming wildness" , "The ...... of the Shrew (Shakespeare)" , "Subjugation" .)