This is Scots and northern English, not much found elsewhere. Its direct sense is that of the cuckoo, that wayward, opportunistic bird who avoids the responsibilities of parenthood by laying its eggs in the nests of other birds. The word goes back to an ancient Germanic source that even then could mean a bastard, simpleton, or fool, laying the ground for its most common sense, that of an awkward or foolish person.
(Though it’s closely similar in form to gawk, to stare stupidly, and gawky, awkward and ungainly, these last two words are from quite different sources. Those who are familiar with the Geordie dialect, around Newcastle-upon-Tyne, will know that gowk means instead an apple core. The two words would seem to have different origins.)
You may have come across gowk in a Robert Burns poem: “Conceited gowk! Puff’d up wi’ windy pride!” Someone who was sent on a gowk’s errand would suffer what those of us from other parts of the world would call a fool’s errand. But its best-known usage connects it with 1 April. Others of us may unwittingly become April Fools, but in Scotland you’re an April gowk.
In particular, you might that day be sent to hunt the gowk, a specialised form of the gowk’s errand. The technique was explained by an author we know only by the initials MTW in a story called April Fools and Other Fools, published in 1881 in a collection entitled Connor Magan’s Luck (well-read readers will spot that it’s an expanded version of the story that appeared in The Book of Days by Robert Chambers, dated 1869):
Having found some unsuspecting person, the individual playing the joke sends him away with a letter to some friend residing two or three miles off, for the professed purpose of asking for some useful information, or requesting a loan of some article, while in reality the letter contains only the words: “This is the first day of April, hunt the gowk another mile.” The person to whom the letter is sent at once catches the idea of the person sending it, and informs the carrier with a very grave face that he is unable to grant his friend the favor asked, but if he will take a second note to Mr. So-and-so, he will get what was wanted. The obliging, yet unsuspecting carrier receives the note, and trudges off to the person designated, only to be treated by him in the same manner; and so he goes from one to another, until some one, taking pity on him, gives him a gentle hint of the trick that has been practiced upon him. A successful affair of this kind will furnish great amusement to an entire neighborhood for a week at a time, during which time the person who has been victimized can hardly show his face.