opendomesday.org lists Green Hammerton with the following entry:

Altera Hanbretone, altera meaning “other”. The entry immediately above on the same folio is listed as being for Kirk Hammerton:

Kirk Hammerton being listed as just Hanbretone.
However, if we look in more detail at the entries, our contention is that these two are the wrong way around on the opendomesday.org site, and the listing for Kirk Hammerton is actually Green Hammerton.
This is the listing for Green Hammerton:
Households
Households: 5 villagers. 1 priest.
Land and resources
Ploughland: 6 ploughlands. 2 lord’s plough teams. 1 men’s plough teams.
Other resources: 1 mill, value 2 shillings. 1 fishery. 1 church.
Valuation
Annual value to lord: 2 pounds 5 shillings in 1086; 4 pounds in 1066.
Owners
Tenant-in-chief in 1086: Osbern of Arques.
Lord in 1086: John .
Lords in 1066: Gamal; Haldor; Thorkil.
And this is the listing for Kirk Hammerton:
Land and resources
Ploughland: 6 ploughlands.
Owners
Tenant-in-chief in 1086: Osbern of Arques.
Lord in 1086: Osbern of Arques.
Lord in 1066: thanes, three.
Our contention is that these two entries are incorrect on opendomesday.org and Green Hammerton is Hanbretone and Kirk Hammerton is Altera Hanbretone. Given Kirk Hammerton would have already had a church by the time of the conquest, is close to the river Nidd for a mill and fishery, this makes more sense than for Green Hammerton.
What the relationship is between the two villages, how they came to be separate, or if they even are separate entities in 1086, is a matter for further research.
Osbern of Arques was a significant land owner in 1086:

There could be more than one person with this name listed in the book, but given the concentrations of locations to the west of York, it’s likely this was the same person.
Finally note the transition from the Norse named landowners (Gamal, Haldor and Thorkil) in 1066 to the clearly Norman named Osbern of Arques twenty years later.