Well, he won't be re-elected in 1844 because his Inaugural Address contained a strong advocacy of a single-term presidency,, an urge that it be made part of the Constitution, and an unequivocal pledge that even if no single-term amendment were adopted, he would not serve a second term:
" I proceed to state in as summary a manner as I can my opinion of the sources of the evils which have been so extensively complained of and the correctives which may be applied. Some of the former are unquestionably to be found in the defects of the
Constitution; others, in my judgment, are attributable to a misconstruction of some of its provisions. Of the former is the eligibility of the same individual to a second term of the Presidency. The sagacious mind of Mr. Jefferson early saw and lamented this error, and attempts have been made, hitherto without success, to apply the amendatory power of the States to its correction. As, however, one mode of correction is in the power of every President, and consequently in mine, it would be useless, and perhaps invidious, to enumerate the evils of which, in the opinion of many of our fellow-citizens, this error of the sages who framed the
Constitution may have been the source and the bitter fruits which we are still to gather from it if it continues to disfigure our system. It may be observed, however, as a general remark, that republics can commit no greater error than to adopt or continue any feature in their systems of government which may be calculated to create or increase the lover of power in the bosoms of those to whom necessity obliges them to commit the management of their affairs; and surely nothing is more likely to produce such a state of mind than the long continuance of an office of high trust. Nothing can be more corrupting, nothing more destructive of all those noble feelings which belong to the character of a devoted republican patriot. When this corrupting passion once takes possession of the human mind, like the love of gold it becomes insatiable. It is the never-dying worm in his bosom, grows with his growth and strengthens with the declining years of its victim. If this is true, it is the part of wisdom for a republic to limit the service of that officer at least to whom she has intrusted the management of her foreign relations, the execution of her laws, and the command of her armies and navies to a period so short as to prevent his forgetting that he is the accountable agent, not the principal; the servant, not the master. Until an amendment of the
Constitution can be effected public opinion may secure the desired object. I give my aid to it by renewing the pledge heretofore given that under no circumstances will I consent to serve a second term."
https://avalon.law.yale.edu/19th_century/harrison.asp
As for other aspects of a Harrison administrastion, as I posted here in 2019:
***
During Harrison's brief administration in OTL there were some signs of tension between Harrison and Clay, which seemed to be working to Webster's advantage. For example, Harrison was annoyed by Clay's insistence that Harrison appoint John Clayton Secretary of the Navy, and in a stormy intervew reportedly reminded Clay that he, not Clay, had been elected president. (Though the man Harrison did ultimately appoint to that office, George Badger of North Carolina, was friendly to Clay). Also, Harrison appointed a Webster ally, Edmund Curtis, collector of the New York Customs House--the most powerful patronage position in the United States--over Clay's vehement opposition. Michael F. Holt,
The Rise and Fall of the American Whig Party, p. 125.
https://books.google.com/books?id=hMkYklGTY1MC&pg=PA125 Moreover, on the issue of whether there should be a speical session of Congress, Harrison at first sided with Webster:
"Significantly, one of the Whigs who continued to oppose a special session was Webster. He realized that he could benefit from Clay's prolonged absence from Washington, and he worried about the disruptive potential of hasty action. Clay, in contrast, passionately supported a special session. When he learned that Harrison, with Webster's concurrence, had tentatively decided against it, he sent Harrison a letter virtually instructing him to call the session and writing out a draft for the presidential proclamation.
"For Harrison, this insulting "lecture" was the last straw. In reply, he chastised Clay for being too "impetuous" and ordered the mortified Kentuckian to communicate with him in the future only by writing rather than through personal visits to the White House. Clay's imperious letter undoubtedly confirmed the president in his intention not to call a special session. When Ewing reported that the government was in debt and faced an additional deficit of more than $11 million unless additional revenue was quickly raised, however, Harrison reluctantly consented. On March 17 he summoned an emergency session of Congress to meet on May 31, 1841, to address the economic crisis.
"Within two weeks of Harrison's inauguration, Clay had fallen out with the new president, and Webster seemed to have won an initial advantage. Both sought to benefit the Whig party as a whole, for only if the party retained its popularity would the nomination in 1844 be valuable. Yet each dearly had a different understanding of what was best for the party and for his own personal ambition. How this conflict might have been resolved had Harrison served out his term can only be speculated..."
https://books.google.com/books?id=5aGyVFn3VnMC&pg=PA127
One should not exaggerate Clay's setbacks: after all, despite Webster being the Secretary of State and despite the absence of Clayton from the Cabinet, it did consist mostly of allies of his (Crittenden, Bell, Ewing, Badger). And Harrison did after all ultimately agree to a special session. But it is clear that Harrison was annoyed with Clay's evident attempts to be the real power in the administration. It is possible that had he lived, Harrison would have worked for Webster as his successor in 1844 (Harrison had pledged to serve only one term). Even if he did so, though, I am unsure he could get the Whig party to go along, and even if they did I am doubtful that Webster, with his image as an elitist and his background as a Federalist, could win the election. But maybe he just has a chance if the Democrats nominate the unpopular Van Buren. (It is not altogether clear that the Democrats would do so, though. True, the Texas issue, which was what allegedly defeated Van Buren in OTL would be less prominent than in OTL--but Texas may just have been an excuse for some of Van Buren's enemies in the Democratic Party. For example, the soft-money, pro-banking faction of the party disliked him and rallied around Cass. And no doubt many Democrats remembered how unpopular his administration had been. In any event, Texas probably does not explain why a considerable number of his nominal supporters
in the North defected on the two-thirds rule.
https://books.google.com/books?id=2vu99nZ2h7cC&pg=PR10)