The phrase "big check" means a substantial amount of money, usually in the form of a large sum of money being given or received for something important or significant. Full definition
It didn't take long for the special interests to start writing big checks. (nypost.com)
Not anymore: All the major car companies have joined the fray — and shown a willingness to cut big checks for the tech behind it. (recode.net)
If such a system would have been implemented, it could have enabled candidates who can't garner big checks from special interests to compete with small dollar donations from constituents. (reinventalbany.org)