To put a hand in and to have a hand in

1) Super power countries usually put their hand in other countries's affairs.

2) Super power countries usually have their hand in other countries's affairs.

Should I use " to put " or " to have"? If I would like to use a adjective to describe the high level of interference , Can I use adj " deep" ? Eg

Super power country puts its deep hand in other country's affairs. ( is it correct? )

Thanks you so much