How can I force my Xiaomi phone to check for a new Google Play system update if it's stuck?

Go to Settings > Security & privacy > System update. Scroll down to the bottom and tap on the 'Google Play system update' option. If it's stuck, you can try to go back and re-select it, or try accessing it through Settings > About phone > MIUI version (for older versions) or System update (for newer versions) and look for the Play system update option there.