Connected regularly at that time (4)
I believe the answer is:
once
'that time' is the definition.
(I have seen 'Single time ' mean 'once' so perhaps 'time' could also mean 'once')
'connected regularly' is the wordplay.
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
The alternate letters of 'connected' are 'ONCE'.
'at' is the link.
(Other definitions for once that I've seen before include "in bygone days" , "'... bitten, twice shy'" , "Formerly; as soon as" , "in bygone era" , "One-time musical film and song" .)