By way of rounding out the source information, here is the title page from Hogg's Relics
The Jacobite Relics of Scotland; being the Songs, Airs, and Legends of the Adherents to the House of Stuart. Collected and Illustrated by James Hogg, Author of "The Queen's Wake", &c &c
Edinburgh: Printed for William Blackwood, Prince's Street; and T Cadell and W. Davies, Strand, London. MDCCCXIX.
"The Bonny Moorhen" was, as Bruce O. noted above, Song LXXVII. It was on pages 129-130. The air was printed in the key or E-minor, but since the air lacks the 6th, it is modally ambiguous. I take advantage of this to harmonize it in the Dorian mode, using for example G-major cords for the IV chord if I sing the air on D.