The phrase "incarcerated fathers" refers to fathers who are currently in jail or prison. It means that these fathers are serving sentences behind bars. Full definition
Frequently, children of incarcerated fathers have found their trust in their father violated and they may tend to project those untrusting feelings on men generally. (thespruce.com)
Supporting incarcerated fathers in the development of healthy marriages and effective fathering skills. (fatherhood.gov)
«Incarcerated Fathers Returning Home to Children and Families: Introduction to the Special Issue and a Primer on Doing Research with Men in Prison» (183 - 200) (fatherhoodinstitute.org)