Do not add a comma in front of the string if value is null or empty
Since you are already combining CONCAT and + to concatenate strings you could just turn them around (although I agree Lennert's solution shows the intent a lot clearer).

Update Companies 
-- set address4 = concat(address4,','+ t.Eircode) 
set address4 = concat(address4 + ',', t.Eircode) 
From companies c
Inner Join Temptbl1 t
on c.comp_id = t.Comp_ID
OR address4 = ''


This works because CONCAT converts NULL into an empty string of type varchar(1), but as long as CONCAT_NULL_YIELDS_NULL is on concatenating a string with NULL will yield NULL.

See this dbfiddle for an example