Your symptoms are more likely due to tonsil stones (tonsilloliths), which can cause white particles, foreign body sensation, and bad breath. This is a common and harmless condition.
However, since you have a history of chewing gum ingestion and now foul-smelling breath, we cannot completely rule out the possibility of a foreign body in the airway or a secondary lung infection (aspiration pneumonia).
Health Tips
Get a repeat chest X-ray with lateral view to check for any retained foreign body or infection.
If cough with fever, breathlessness, or foul sputum continues, consult a pulmonologist for evaluation and possible bronchoscopy.
Meanwhile, continue good oral hygiene and warm saline gargles.