For an argument that Clay could indeed have avoided a war, see Gary Kornblith's "Rethinking the Coming of the Civil War: A Counterfactual Exercise"
https://web.archive.org/web/2017082...te/assets/documents/02_JAH_2003_kornblith.pdf
"The key to peace in my counterfactual scenario is a victory by Henry Clay over James K. Polk in the very close presidential election of 1844. Had Clay won 5,107 more votes in New York State (out of more than 485,000 cast there), he would have become president. As a result Texas would almost certainly have remained an independent republic in 1845, and the United States would not have gone to war with Mexico the following year. Based as it is on a highly plausible turn of events, this counterfactual scenario promises to advance our understanding of thecauses of the Civil War...
"Having established that a Clay victory in 1844 was thoroughly plausible, we can, proceeding with our thought experiment, consider the implications of such a victory for American political and social development. First, we must ponder the fate of Texas. Unable to point to the electoral outcome as an endorsement of his policies, Tyler would not have succeeded in pushing through Congress a joint resolution authorizing annexation before he left office. At the urging of the president-elect, Whigs in the Senate—southern as well as northern—would have cited Clay’s victory to justify their continued opposition to annexation. Had Henry Clay taken office as president on March 4, 1845, he would have enjoyed a good deal of flexibility in crafting his policy toward the Lone Star Republic...
"In 1843 George Gordon, Lord Aberdeen, the British foreign secretary, floated a proposal coupling Mexican recognition of Texan independence with abolition and British assistance. Sam Houston, then president of the Texas republic, briefly displayed some interest. But Aberdeen withdrew the suggestion of a quid pro quo the following year, and thereafter he temporized on the question of Texan abolition for fear of encouraging American annexation. Given competing foreign policy priorities, the British were unprepared to risk major involvement in Texas in the face of American opposition. A President Clay would have objected to any British effort to promote abolition in Texas for the same reason he opposed annexation of Texas: his overriding concern was the maintenance of sectional harmony and American political stability.
"In all likelihood, Aberdeen and Clay would have joined diplomatic forces in support of Texan sovereignty. In early 1845 the British and French undertook a new initiative to convince Mexico to recognize Texan independence, and Mexican authorities reluctantly agreed. Although the Mexican government would have felt less compelled to comply had Clay rather than Polk been elected, it could not have comfortably ignored the combined pressure of Great Britain, France, and the United States. We may postulate that sooner or later during Clay’s presidency Mexico would have recognized Texan independence and entered into international arbitration over boundary issues. Even had the Mexican government continued to refuse official recognition, it would probably have shrunk from open warfare and allowed the Lone Star Republic to consolidate authority and power further..."
As for California, Kornblith writes:
"The destiny of California under a Clay presidency is harder to project with confidence. Even without the advent of war with the United States, Mexico would have sustained its claims to sovereignty only with difficulty. In early 1844 the beleaguered governor of California, Manuel Micheltorena, recommended to his superiors in Mexico City that they consider handing the province over to British creditors rather than let it fall into the hands of American immigrants and californios (Californians of Hispanic descent). 'In August 1844,' wrote David J. Weber, 'a group of californios met secretly with British vice consul James Forbes in Monterey and told him they were ready to drive Micheltorena out of California, declare independence, and ask for British protection.' Without instructions from London, Forbes was stymied, but the rebels nonetheless succeeded in ousting Micheltorena in early 1845. They stopped short of declaring independence, however, and soon divided among themselves. Meanwhile, Americans in California prepared to take matters into their own hands, and in June 1846 they staged the Bear Flag Revolt. 'Even if [the Mexican-American War] had not occurred,' Weber asserted, 'Americans in California had become numerous enough to think they could play the 'Texas game' and win.'
"Whether the discovery of gold in 1848 would have prompted President Clay to show more enthusiasm for annexing California than he did for annexing Texas is hard to know. Fellow Whig (but political rival) Daniel Webster had long hoped to acquire San Francisco and the surrounding area for the United States. Yet Clay was more sensitive than Webster to sectional tensions and to the explosive consequences of adding new territory to the federal domain. As with Texas, Clay might well have preferred strong commercial ties with California to the national and international controversies sparked by annexation. For this reason, he would probably have encouraged California to remain independent so long as it avoided an open alliance with Great Britain or another foreign power. Certainly, the possibility that California could have flourished as a separate nation deserves serious consideration. The historical geographer D.W. Meinig has written, 'Was there ever a region better designed by Nature for separate geopolitical existence than Alta California--a land so distinctive and attractive, set apart by the great unbroken wall of the Sierra Nevada backed by desert wastelands, fronting on the world's greatest ocean, focused on one of the world's most magnificent harbors?'
"Alternatively, under pressure from Democratic expansionists in Congress, a President Clay might have proposed pairing the annexation of Texas and California--a reprise of the Missouri Compromise with its coupling of Missouri and Maine. But that scenario seems less probable than the establishment of an independent California because it presupposes Mexico's peaceful acquiescence, a most unlikely development. In keeping with past policy, Britain would have supported Mexican objections to American annexation (as distinct from Texan or Californian independence) and Clay would have backed away from a war for territorial expansion. His commitment to diplomacy, rather than force of arms, would almost surely have curtailed the country's westward growth for the duration of his presidency..."
Kornblith argues that with Clay as president and no Mexican War, US political parties would continue to divide on economic issues (banks, tariffs, etc.) with the slavery issue muted. The whole Wilmot Proviso controversy of OTL would not take place, and it was this controversy which gave birth to the Compromise of 1850 with the application of "popular sovereignty" to the newly acquired southwestern territories. It was this same doctrine of "popular sovereignty" that was applied to Kansas-Nebraska in 1854, with fatal results to the second party system, and the emergence of a new party, the Republicans, whose victory would lead to southern secession.
(There are all sorts of problems with this scenario, as Kornblith recognizes. For one thing, the expansionist Democrats, having lost in 1844, might win in 1848...)