Baby 1st month

My baby turned 1 month this June 28, 2020. He is due for a vaccine but he still not getting well from his phlegm sa throat di mawala. Wala na siya sipon, only mucus or phlegm pag ubo or after breast feet. We gave him Salinase already. Dapat na ba siya vaccine or pagaling muna ng ubo? Thank you po sa mga magreply.