A deep cough sometimes with phlegm no other cold signs. Gets worse at night?

Night time cough. Coughing at night could be due to a number of conditions, including acid reflux, sleep apnea, asthma, and heart problems. A primary care physician would be able to help determine the cause.
Could be. Several different things. Sometimes infections are very low grade and you may just be stricken by an annoying cough. But this could also be related to acid reflux or sinus issues complicated by post nasal drip(allergic or not). If you find it bothersome seek an in person evaluation. .
Allergies/ asthma? These symptoms could represent allergy symptoms or possibly asthma since the symptoms are worse at night. I would suggest trying an over the counter antihistamine such as Zyrtec (cetirizine) at bedtime x one month. If your symptoms do not improve by then, I suggest seeing your Doctor for an evaluation.