Is there a relationship between female genital cutting (fgc) and fistula?

Yes. There is a significant risk for women with type iii female genital cutting to developed vesicovaginal & rectovaginal fistulae.
FGC is barbaric. Only performed in backward and primitive communities in middle east and northern africa. Clitoris is amputated with a sharp knife and urethral meatus may also be amputated. Worst fgc is wehere the vagina is sewn almost shut. Procedure is not performed by skilled surgeons (who should have their genitalia amputated if them perform such mutilation) and suturing vagina can certainly cause a fistula.
Yes. It is possible, since cutting takes place during childhood the organs are still developing and it is possible to develop fistula if the wound did not heal appropriately.