Ex–British Prime Minister has egg and a thick cold sauce (4)
I believe the answer is:
mayo
'a thick cold sauce' is the definition.
'mayo' can be an answer for 'sauce'. I'm unsure of the 'a thick cold' bit.
'ex-british prime minister has egg' is the wordplay.
'ex british prime minister has' becomes 'may' (I can't justify this - if you can you should give a lot more credence to this answer).
'egg' becomes 'o' (I've seen this in other clues).
'may'+'o'='MAYO'
'and' acts as a link.
Can you help me to learn more?
(Other definitions for mayo that I've seen before include "Salad dressing (informal)" , "Eg, Braeburn or Cox" , "Ir. county; salad dressing" , "radio presenter" , "County of Western Eire" .)