There was a ship, the ABERCORN, left Londonderry, Ireland for Wilmington with lots of youngsters on it, and others in their twenties - and I have a feeling that this might have been to do with the Potato Famine in Ireland at about that time. I am only guessing because I don't know the true dates of the famine, just that it was around that time.
The ABERCORN had 471 passengers, and among them was a Robert POLK aged 5 and with him a John POLK aged 6.
....However, they arrived 28 May 1840 and I think you were looking for a voyage around 20 years earlier, so that is a drawback. ... I had a look at the first page of the Passenger List, and it seems to be saying that the full name of the ship was the MARCHIONESS of ABERCORN, although it isn't clear to make out.
...If you can't find "UK to Atlantic Ports" on Ancestry, you can get the passenger list by googling "Ancestry - Ships Passenger Lists - UK to Atlantic Ports 1820-1873." When the page comes up, type in surname POLK. Birth: Ireland. Departed: Ireland. Ship ABERCORN. Gender Male. Race: Irish.
I think you have to take into account that there were lots of mis-spellings in those days, and POLK could be POLLOCK.
It probably isn't them but it is the nearest match I could find,