Hi Buzmar,
sorry to hear about your father. Yes, your father can take flu shot as it is not a live virus but take an advice from your oncologist.
why don't you ask your oncologist about G-CSF which stands for Granulocyte Colony Stimulating Factor.
My mom is on Gemzar and Cisplatin and her WBC count went too low to give her second cycle of chemo. Since then she has been given G-CSF under her skin twice in one cycle. Her WBC count goes to 24,000 after jsut single injection which comes back to around 5000 (which is normal) just prior to her next cycle.
So talk to your oncologist regarding G-CSF. Your father doesn't have to worry at all for infection or treatment delay once he is on G-CSF.