Hi welcome to the health care magic Here speckled cough discharge suggest towards upper respiratory tract infection like sinusitis or bronchitis I need history about whether presence of chest pain, dyspnea, fever etc... Your work up done as follow.. _Auscultation _PNS x ray _if needed chest x ray _CBC According to report further treatment can be guided Until than take decongestant like levocetrizine or phenylephrine Drink more water to loosen mucus esp warm water Steam inhalation done three times a day You can take Ambroxol containing syrup or tablet for mucolytic effect Until you do above prescribed report, no need of starting antibiotic Hope your concern solved