I would firstly check the photocell is clean, if you haven't already. Highly likely that is the problem, but the nozzle would need to be clean and okay and oil/air set up properly or perhaps the burner would fail to cold start easily.
Although I wouldn't rule out control box fault, I would think the oil pump could have a fault and slipping.
In short look for these until you find which is faulty, - (1) clean the cell, (2) flame okay? (3) oil pump. If all else fails then control box has a slim chance of being faulty
Coil would not be faulty as it has to work continuously and it obvious must be working if burner eventually fires and keeps going.
The solenoid valve the coil operates however, on the oil pump could stick a bit.