No, it's neither correct nor natural. There's no subject, which makes it incorrect. We do that sometimes, but in this case it's not natural either. The use of 'have got' is also rather strange, as you point out. Normally we would just say 'got' or 'have'.

have got = have Sometimes I just have got a sudden idea to go shopping= Sometimes I just have a sudden feeling that I would like to go shopping.