N.H. Rev. Stat. § 657:16
If he or she refuses to certify the application, the town or city clerk shall notify the applicant in writing within 7 days to that effect. The town or city clerk shall provide the applicant with an absentee ballot and a notice that the ballot will not be counted unless the applicant submits the documents necessary to complete an absentee registration. The applicant shall be advised in writing what documents, if any, have been received in proper form and which the applicant must submit in the outer envelope that contains the absentee ballot envelope. The town or city clerk shall mark the absentee ballot application, the absentee ballot affidavit, and the outer envelope with the words "Not Registered." Upon receipt of an outer envelope marked "Not Registered," the clerk shall open the outer envelope. If the applicant returns the required documents with the absentee ballot by the date set for correcting the checklist under RSA 654:27 and RSA 654:28, the town or city clerk shall forward the registration forms to the supervisors of the checklist and, if the applicant is found to be qualified, the applicant shall be registered and his or her absentee ballot shall be processed in the same manner as the absentee ballot of a previously registered voter. All documents received after the deadline for correcting the checklist under RSA 654:27 and RSA 654:28 shall be processed as election day registrations under RSA 654:7-a. If the ballot is returned without the required documents in proper form, the ballot shall be marked in the manner set forth by law for successfully challenged absentee ballots and preserved in accordance with RSA 33-A:3-a. The clerk shall preserve the application of any applicant who is not registered as a voter until the time set by law for the destruction of the ballots after the election at which time the application shall be destroyed. Any justice of the superior court has jurisdiction in equity upon such notice as he or she may order to require that the name of the person making application for an absentee ballot be placed upon the checklist or registered as a member of any party and be sent an absentee ballot.
RSA 657:16
1979, 436:1. 2003, 289:54. 2010, 317:79. 2014, 74:1, eff. July 26, 2014. 2016, 317:2, eff. Aug. 23, 2016.