Van Buren was unpopular due to the Panic of 1837, so Clay probably wins.
And the impact on the Whig Party in the longer term?
Hard to say. At the time, after Jackson’s “imperial presidency,” many Whigs were big on the idea that presidents shouldn’t serve more than one term; this is why Polk pledged to only serve one term in 1844, to undercut them on the issue. If Clay runs again in 1844, Polk could use that against him. If he doesn’t, and someone else is the nominee, hard to say what happens. But I think probably the most likely scenario is Polk wins in 1844 anyway, so not sure if there’s a huge long-term change for the Whigs.
Still one thing they have going for them with a President Henry Clay is that they can actually enact their
American System agenda with a protective tariff, a national Bank of the United States, and massive federal subsidies for infrastructure development. Something which never happened in OTL with President John Tyler (a Whig-in-name-only) who took over upon the untimely death of his predecessor.
I don't know how popular this agenda would be, but it could help people gloss over whatever 'betrayal' President Clay would be committing in deciding to run for re-election in 1844.