Flu shots might give some protection (the adjuvant gets the immune system going) but that would only be transient and not very effective. The flu is a very different family of virus and, generally, i would not expect much for cross protective immunity from a flu shot. A covid vaccine is specifically targeted to get your body to protect yourself against a strain of covid. If we could target a trivalent 24.D XDV.1 / 25I BA.3.2 / JN.1 that would be ideal. But I don't think the next round of vaccines will do that [we'll see]