I thought you might like to read this, taken from James C Dick's book 'The Songs of Robert Burns and Notes on Scottish Song by Robert Burns':

In the Commonplace Book, entitled 'Har'ste:- a fragment, are eight lines substantially the same as begins the song which the sister of Burns said was written for Jean Armour. The complete song is in the Kilmarnock edition 1786, 224, entitled 'Song, composed in August'. Tune,'I had a Horse, I had naer mair', and the MS. is in the British Museum. Burns changed the heroine to Peggy Thomson, who lived next door to the Kirkoswald school, where Burns studied trigonometry, and she 'upset all my sines and co-sines, and it was in vain to to think of doing any more good at school'. She subsequently married a Mr. Neilson, and Burns was on friendly terms with both. When the song was revised, Burns altered the melody to Port Gordon, as may be seen in the Gray and Law MS Lists, but Johnson of the (Scots Musical) Museum neglected the instruction, and attached the melody When the King Comes o'er the Water, titling it erroneously Come, Kiss with Me. Thomson, in Scottish Airs, 1799, 93, mutilated the verses, and adapted them to an Irish air 'Ally Croker'. The tune Port Gordon, for which Burns wrote the song, is in the Caledonian Pocket Companion, c. 1756, viii. 25. There is a familiar resemblance, but the air is not the same as When the King Comes o'er the Water.

I have listened to Dick Gaughan and various other singers on YouTube but not one of them sings either the 'Come, Kiss with me', 'Port Gordon' or 'I had a Horse, I had nae mair' tune. I wonder if they are singing the 'Ally Croker' tune which I don't have at the moment, or did someone originally 'invent' their own tune and everyone else copied it?