Due to the date, the 1852 marriage would probably be non-Catholic, unless it took place in a registry office - so possibly Church of Ireland.
The Representative Church Body Library in Co. Dublin holds many of the historic CofI records, and they have details for the parish of Youghal back as far as 1666. As agho mentioned these are not available online for this parish, you would have to visit the RCB library to view them.
You could establish a few additional details that might help, by ordering the marriage cert for John and Elizabeth. e.g. this would show the name of the Church, and the denomination. Occupations for the groom and the two fathers, names of two witnesses, and also current residences for the bride and groom (although these can be basic). These are the details you need to order the cert from the GRO :
Name: John Hammett
Registration district: Middleton
Event type: Marriage
Year: 1852
Volume : 8 / Page : 78
Name: Elizabeth Hannan
[same references as John]
see :
Ordering Certs from GRO Roscommon Details included on a Marriage CertShane
p.s. not sure why the civil marriage was listed in the district of Midleton and yet the extracted record states Youghal.. but the districts are adjacent.