For comparison, although the UK
sometimes produces a two-party coalition government (including twice very recently) it's more common for there to be a single party with a parliamentary majority.
In which case essentially the only substantial "check" on the prime minister's power is the next election. They don't need to buy the support of anyone from another party - and they can count on the support of their own party.
Of course, UK governments will still spend lots of cash in marginal constituencies in hopes of buying the locals' votes. And there's still corruption, like ministers handing out lucrative PPE contracts to their personal friends; handing out honours in exchange for donations to their party; and retiring to highly paid do-nothing jobs in industries they previously regulated.