The videos were understood to have been filmed at a Duke St property.
Hodgson and a few teammates were at the party, having been invited into the flat - opposite the hotel in which the English team were staying - by celebrating students.
The players are understood to have arrived at the party about 4am, after a few hours in town, and left after 8am.
The team were in Dunedin for their final round-robin game of the 2014 Four Nations tournament against the Kiwis - a match NZ won.
England manager Jon Roberts said he was aware of the incident.
"It is being dealt with internally with the person involved and they will be disciplined."
The players had been given permission to go out after Saturday's game, he said.
Watch: England league star's smashing entrance Part 2
"On behalf of England Rugby League, we apologise for the behaviour and have arranged for the repair of the small amount of damage.
"We have had a great time in Dunedin and the city has made us welcome. [Saturday's] match was a great spectacle for international rugby league and we look forward to returning to the city in the future," Mr Roberts said.
A Four Nations spokesman said the tournament had been a success.
"The incident in Dunedin is thus disappointing. We have been in contact with English officials, who are dealing with the matter."
Dunedin police had not received complaints about the incident.
On Tuesday, the team went to the Wingatui races in Dunedin, but had to leave because of the antics of a group of drunk students.
Hooker Daryl Clark told the Daily Star he had never seen as many drunk people in his life. "We were all sober so it was a bit of an eye-opener. We didn't stay too long when we realised what it was like."
- additional reporting: Otago Daily Times
