Here's the version from the Oxford Books of Sea Songs. Very similar to the lyrics posted by JeffB in the first message, but with significant differences. I contacted JeffB, and he did not know the source of his version.

CAPTAIN MANSFIELD'S FIGHT WITH THE TURKS AT SEA

Our goodly ship was loaded deep,
With anchors three beneath her bow;
'Twas east-north-east we steered our course,
And as near the wind as we could stow.
We had not sailed glasses three,
Nor yet ten leagues from loaden port,
Before we spied ten Turkish men-of-war,
And after us they did resort.

'O hail, O hail, you English dogs.
O hail, and strike your sails quickly,
For you shall go with us this night,
And ever after into slavery.'
O then bespoke the captain bold,
And a well-bespoken man was he:
'If you must have my topsails down,
Come on board and strike them for me.

'To the top, to the top, my merry boatswain,
To the main topmast-head so high,
And see your business you supply.
To the top, to the top, my boatswain's mate,
To the fore topmast-head with speed,
And sling me here the fore topsail yard,
For we never had any more need.

'To the top, to the top, my little cabin boy,
To the mizen topmast-head so high,
And spread abroad St George's flag,
For by that we live or die.'
O then bespoke our gunner bold,
And a well-bespoken man was he:
'Swab your guns, brave boys, while they're hot,
For powder and ball you shall have free.'

'Keep aluff, keep aluff,' says the master's mate,
'Keep aluff whilst that you may;
We'll fight it out like English boys;
It never shall be said we run away.'
So to it we went like lions bold,
As enemies do when they meet;
From twelve o'clock to sun rising
We spied but one sail of their fleet.

O three we burnt and three we sunk,
And the other three run away;
And one we brought to old England
To show we had won the day.
All you that know our gallant ship
And want to know our captain's name,
It is Captain Mansfield of Bristol town,
And the Marigold a ship of fame.


Notes: In December 1669, Captain John Kempthorne in the warship, Mary Rose, was attacked by seven Algerian vessels, but managed to repulse them, after a fierce action lasting four hours, and make his way to Cadiz. On returning to England, Kempthorne was knighted for his efforts, a signal honour for a 'tarpaulin', an officer who had first risen to command in the merchant navy. He took his son, Morgan, to the ceremony, and Charles II said: 'Bring him up to the sea. We desire more of the breed.' Morgan duly went to sea, and was killed in 1681 when commanding the Kingfisher in a battle with five Algerian ships off Leghorn. Wenceslas Hollar engraved the father's battle, and Van der Velde the son's. It seems likely that the ballad was inspired by the father's battle, and updated for the son's, though no contemporary printing of either has survived. The version given here dates from the eighteenth century, which perhaps explains why the names have been changed. Oral versions from landsmen and sailors of recent times show further variations, but the essential theme of British grit remains.

Source: The Oxford Book of Sea Songs, chosen and edited by Roy Palmer (1986) - #19, page 41.