Is it possible to pass a wart from your hand onto your penis?

Yes

Warts (http://www.kidshealth.org/teen/infections/bacterial_viral/warts.html)

Does that mean it is genital warts the sti or summat else?

Warts are a virus. They can and do spread. All warts are a form of the herpes virus.

If they are on your genitals they can be spread to your partner's genitals.