A bit more from the Hal an Tow song also gives some historical background and points to the time of the Chanty's origin:

We were up long before the day-O
To welcome in the summer,
To welcome in the May-O;
For summer is a comin'in,
And winter's gone away-O!

May was when ships put out for long trips - the spring tides and summer winds made travel more pleasant - The sailors really haven't gone a'Maying, but they are enjoying the fair winds and following seas

What 'appened to the Spaniards
That made so great a boast-O?
They shall eat the feathered goose,
And we shall eat the roast-O...

This verse is a reference to the battle with the Spainish Armada, the Spaniards boasted that they would defeat England and take it over, since Elizabeth refused to marry their prince.

The sailors, who have just enjoyed the Spainish defeat declare that the Spainish will only eat arrows which have goose feathers on them
for stabalization (fletching)

But the victors get to eat the bird which provided the arrows - this suggest that this is a song dating back to the war with the Spainish and a time when arrows were still used as weapons aboard ship. It also suggests that the English sailors were not as well armed as the spainish.

Robin Hood and Little John,
They've both gone to the fair-O.
And we will to the merry green wood
To hunt the buck and hare-O...

This may be from an older song, or it may refer to a story about Robin hoods good shooting - again the sailors are saying they sent accurate arrows against the Spainish

God bless Aunt Mary Moses
In all her power and might-O,
And send us peace to England,
Send peace both day and night-O...

The first line is a bit garbled but it blesses both the Virgin Mary for her help with the war
And it wishes a blessing on Moses, in this case representing the Jewish merchants who were funding the ship's expedition - this blessing on Moses occurs in other sea songs.

Other version include something that delights pagans - references to wearing deer horns, skins, and the deer head crest.   The stags head crest may have been an old symbol for England,just as is the English Lion, and Unicorn are symbols, or it may have represented the ship owners or captains crest.

There is references to it as belonging to your father and grand father but, this version also includes a rude joke about how having horns means that your wife is committing adultry - a sailor on an extended trip would worry about this - and the joke would be don't worry, that's how you and your whole sea faring family came to be!