Yesterday started badly, with a sore eye (4)
I believe the answer is:
stye
'yesterday' is the definition.
Although both the answer and definition are singular nouns, I cannot see how they can define each other.
'started badly with a sore eye' is the wordplay.
'started badly' becomes 'sty' (I can't explain this - if you can you should believe this answer much more).
'with' is a charade indicator (letters next to each other).
'a sore' becomes 'red' (red can mean sore or painful).
'eye' says to take the centre (eye can mean the centre of something).
The middle of 'red' is 'e'.
'sty'+'e'='STYE'
Can you help me to learn more?
(Other definitions for stye that I've seen before include "Observer's problem" , "Eye sore" , "Eyelid infection" , "Eye problem" , "Swelling near the eye" .)