Monmouthshire Poor Law Unions/Workhouses (Wales) is a free site at FamilySearch Wiki with a list of towns in Monmouthshire with workhouses or Poor law unions and where they were located.
Workhouses were traditionally places where the working poor and those unable to sufficiently support themselves were offered a place to live and work in exchange for food and shelter. By the end of the 19th century, they hosted many of the elderly and infirm instead of less fortunate working people.
Also visit Workhouses.org.uk, a free site authored by Peter Higginbotham with records from Workhouses across the UK including England, Ireland, Scotland, and Wales. Watch “The History of the Workhouse with Peter Higginbotham” at FindMyPast for a history of the Workhouses.