Ah yes, I was maybe too hasty to change the spelling, if newer sources say so. Older historical sources (OS 25 Inch, 1892-1914) and this OS name book (https://scotlandsplaces.gov.uk/digital-volumes/ordnance-survey-name-books/fife-and-kinross-shire-os-name-books-1853-1855/fife-and-kinross-shire-volume-74/68) from the 19th century use "Clinkhill".

I did also, come across a recent story (https://www.heraldscotland.com/news/13097547.Police_find_body_in_missing_jogger_hunt/) where a "Clinthill" is mentioned, so fair, I'll revert my rename, and maybe verify by survey sometime I'm in the area.