I dont know any details on surviving Workhouse records for Drogheda, but presume that the local library would be the place to check first.
There's a possible marriage in the right area, and at around the right time, on the Civil BMD Index for Catherine :
Name: Catherine Weldon
Registration District: Ardee
Event Type: Marriage
Year: 1870
Volume : 7 / page : 373
(The registration district of Ardee is based in Co. Louth, and not far from Drogheda )
I dont see any cross-match to Thomas, but maybe his name was mis-transcribed. Given that the name Catherine Weldon is quite unusual, I'd say there's a reasonable chance that she's
your Catherine.
Based on the extracted record details that aghadowey mentioned for the children - these are the references required to order a cert for Laurence, in case you wish to order a cert to see the full details :
Name: Laurence Dignum
Registration District: Drogheda
Event Type: Birth
Year: 1876
Volume : 2 / page : 558
see the topic
Introduction to Irish Records for details on searching the BMD Index, ordering certs etc
Shane