Later today, that man's name covered up by drunk (4,5)

'later today' is the definition.

'man's name covered up by drunk' is the wordplay.
'man's' becomes 'his'.
'name' becomes 'n' (common abbreviation e.g. on forms).
'covered up by' indicates putting letters inside.
'drunk' becomes 'tight' (I've seen this before).
'his'+'n'='hisn'
'hisn' inserted inside 'tight' is 'THIS NIGHT'.

'that' is the link.