It could be due to erosion in throat .If you cough
hard then it can damage the mucosal membrane of throat and resulting in blood in sputum which exactly has happened with you.
you can take drosera 30 4 tabs 4 times in day for 3 days and report afterwards.
If blood in sputum increases you can take phosphorous 30 1 dose of 4 tabs and then also if it does not stop consult the doctor immediately.
Your problem will be resolved by the medicine,
hope this helps,