Horse’s messy saddle put into cold washing solution (10)
I believe the answer is:
clydesdale
'horse's' is the definition.
(Clydesdale is an example)
'messy saddle put into cold washing solution' is the wordplay.
'messy' is an anagram indicator.
'put into' means one lot of letters goes inside another.
'cold' becomes 'c' (eg on taps).
'washing solution' becomes 'lye' (I've seen this before).
'saddle' with letters rearranged gives 'desdal'.
'c'+'lye'='clye'
'desdal' inserted inside 'clye' is 'CLYDESDALE'.
(Other definitions for clydesdale that I've seen before include "Breed of carthorse" , "Powerful breed of horse" , "Draught-horse" , "Heavy breed of draft horse from Scotland" .)