Here’s a link to a website with some information on Cookstown Workhouse.
It doesn’t look as though the Cookstown workhouse admission records for 1870 have survived. (They only appear to start in 1899, and they are the records you really need). There are the Board of Guardians minutes (weekly management meetings) for the period you require but they usually only name inmates if something significant had happened to them. However they do give a flavour of life in the area at the time. They are in PRONI in Belfast if you want to read them. They aren’t on-line.
Though the workhouse was obviously pretty grim people were admitted and discharged all the time. It wasn’t prison and if they could support themselves people often left. Many children were adopted or sent to Australia etc on paupers emigration schemes. The Workhouse infirmary was – prior to the introduction of the NHS in 1948 – pretty well the only free hospital care available in the country, so some people went there for medical treatment rather than because they had nowhere else to stay.
Formal adoption was only introduced in the UK (including Northern Ireland) in 1927 or thereabouts, so in 1870 any adoption was an informal process arranged by parents, grandparents, the church or various other worthy bodies. No paperwork was necessary and it’s fairly unusual to find any record of such arrangements from that period.