Could a virus linger in the body for a few months and then after that have terrible flu?

I had a bad case of flu a few weeks back. I had picked up an irritating cough from a child I looked after and she was sick although I didn't get sick until 2 months later. I wasn't around anyone in the days before I got sick and now that the flu is over I am still getting an irritating/chesty cough and thick bloody boogers up my nose (tmi sorry).