Hi Karen,
The country was divided into Poor Law Unions --- but I think John Wilkins was just in the Infirmary as a patient --- it doesn't say 'inmate'. (Many Infirmaries were part of a Workhouse -- and if they were admitted simply because they were in dire financial straits -- they would be described as an 'inmate'.
This may just be a coincidence -- but a John Wilkins aged 49 died in Greenwich Union Infirmary on 9th May 1922 - due to Tuberculosis/Phthisis. Home address at that time was 17 Barnes Terrace, St Pauls. (This record amongst the London deaths on Ancestry). He is about the correct age.
I don't know if this is your John -- as there is no other information.
If you think about it --- it was extremely difficult for a widower to hold down a job AND look after young children. Many remarried very quickly to solve the problem. If John didn't remarry -- the children were probably taken in by the relative just for convenience sake -- he would need someone to care for them.