I'll answer this best I can. I welcome any American historians to chip in if they can answer too.

If we still named people "the Something" as we did centuries ago, Henry Clay would undoubtedly be "Henry Clay the Compromiser". He was notorious for making compromises to positions he had held for long periods in order to gain something for himself. In 1889, John T. Morse wrote that "his panacea, his watchword, his purpose was always Compromise". Peter Knupfer says that his only "two firm, consistent opinions" were his "hatred of Andrew Jackson and his patriotism." As you noted in your title, most of the time his goal was the Presidency. In pursuit of this, he compromised his policies repeatedly.

I can find two major examples which I think qualify to answer your question: the rejection of the emancipation of the slaves from an original position of complete emancipation; and the abandonment of his key platform of policy which he called the "American System", and which he used for his first two presidential campaigns.

For anyone else who is new to Henry Clay's story, he ran for President in 1824, 1832, and 1844. He was from Kentucky, was the Speaker of the House three times, and was a one-time Secretary of State. All in all, he was quite qualified.

Of the two compromises I pointed out which I think would count as major changes to his platform, the American System was the one which was more personal to Clay. The American System was an economic policy that established a central banking system(early vision of the Federal Reserve, which I believe was in existence as the Second Bank of the United States), and provided tariffs to protect American manufacturers. By the way, these tariffs would significantly aid Clay's personal business if they came to fruition. The American System was a key part of Clay's first two presidential campaigns. By 1844 however, he had dropped it from his platform as it cost him the previous election, when he was painted as a capitalist conspiring against the working classes.

With his first two elections lost while running on a platform of emancipation and the American System, and the American System dropped soon after the 1832 defeat, Clay also turned around and changed his views on slavery. Previously, Clay had supported the full emancipation of all slaves, and the compulsory emigration of all freed slaves back to Africa. By 1839, with two failed campaigns behind him, and the American System dropped, he found himself speaking in the Senate against emancipation, as his sole path to the White House now lay in the hands of the Southern states - in other words, the ones where slavery was viewed as an economic necessity. Whether his personal views on slavery changed, I do not know. What is clear though is that by this point, at the age of 62, Clay had realised he could not win the presidency without the support of the slave states, and so he shifted his political platform to suit.