Is this hemophilus infection?

So I had viral infection for 10 days which are mostly gone and improving on day 11, but all tests for viral infection PCR came negative for the common viruses, the only thing that they found so far is hemophilus influenzae culture in my nose, but not in the throat swab.

Should I do cough test?

The symptom the general (fever, fatigue) and local (lowe throat pain, sinus pain, nasal congestion) are nearly gone and improving.

The last blood work came fine except the iron is a bit lower than the reference and granulocytes, the CRP it was 50 on day 6, on day 9 it went down to 13.2 mg/L.

PCP right now preacribed me chloramphenicol antibiotic cream in the nose 3 times a day, I've used it 2 days since starting it.

The current symptoms are only occasional sinus pain in the upper sinuses near the forhead, occasionally have fatigue, and the temp varied near 37.3-37.4 when had fatigue then goes to 36.2-36.7. The sinus pain went to headache yesterday and I took ibuprofen.

Should I proceed for couple of days with the cream? Or should I push for systemic antibiotic?

Is it possible if hemophilus is not the cause of the symptoms now to be actived with antibiotics somehow by disruption of the nasal flora, or it does not work like that?

Any suggsstions?