Yes, Overseers records, Guardians' Minute Books and suchlike - they tend to be called different things in different areas.
Overseers papers tend to refer to poor relief pre-1834 - because every parish appointed one or more 'overseers of the poor' to manage their statutory requirement to look after the poor of their parish.
The Poor Law Amendment Act of 1834 changed the system. Groups of parishes were formed into Unions and a Board of Guardians was set up in each union to handle poor relief for all the parishes in the Union. So Parish Overseers had a lot less to do post-1834.
This is why the Berks FHS CD of Overseers Papers ends at 1834. It includes a calendar (ie a precis) of all known overseers papers that survive in Berkshire (in practice this means all such records held in the BRO - some more may still exist at a few churches). In reality there are a few that were missed - I have seen some entries in the Newbury Borough Quarter Sessions records that should really have been added. That said the book of Newbury settlement examinations is one of the best bits of the CD!
Anyone who is receiving parish relief at home in 1851 is probably not able bodied (or they would be in the Workhouse). Workhouses did take the infirm if there was no family to look after them but their main intake were the able-bodied poor, who were given such a miserable time that they would be desperate to find work and escape the workhouse (which was exactly the idea as working folk were not a drain on the parish rate). Looking after the infirm in the workhouse cost money and had no secondary effect as they were not expected to rush off and find a job - so out-relief could be a cheaper alternative.
Over the years workhouses did metamorphise into hospitals, but 1851 is too early for that to have happened to any great extent. In 1929 most surviving workhouses were formally converted into hospitals (eg Battle Hospital in Reading & Sandleford Hospital in Newbury). Bradfield Workhouse was out in the sticks but even that became Waylands Hospital.