But I don't think the song "When my rowboat comes in" changes key between verse and refrain. I still think the chord in 'sun' is just a good old A minor.

The normal chords for F are

F Gm Am B-flat C Dm

and I bet they would work.

If Steve had flirted at some point with 'the belly-dance minor,' then he could have played an A. If you get stuck, try an A.