In comparing additional records to the previous answer, this may confirm the James and Hannah. On several of these records - the Street Address for Thomas matches the street addresses for James and Hannah!making it more likely that Hannah T Mahoney is married to the correct James. And the mother is again Mary Harkins.It appears that Michael and Mary Harkins had 10 children but only 4 were living in 1900.You can probably find more of the childrens records on familysearch - paying attention to Street - if given.
Mary Harkins Lynch and Thomas are also buried at Holy Cross cemetery, Malden - as were Hannah T Mahoney Lynch and her child.That might be an addtional place to contact.
All from found using the FREE search tool at:https://familysearch.org/
If you are not familiar with familysearch - click on any name in the first column in the results list (if there is a camera icon) - then that means there is also an IMAGE of the record.Once you click on the name, it opens an additional page.On that page - on the upper left - it should say VIEW IMAGE - click on that to see the actual record.Some images, such as some census years - are pay sites like ancestry.com - but many other images are FREE - such as the birth, marriage and death records.
U S Census, 1900 ED 1207 Precinct 6 Boston city Ward 4, Suffolk, MA headMichael LynchM 62 est b Apr 1838 IREimmig abt 1857 marr wifeMary Lynch F 57 est b. 1843 IRE10 children 4 living sonThomas Lynch M 34 est b. Aug 1866 MA sonJames Lynch M 25 est b. Feb 1875 MA
Massachusetts, Deaths name: Thomas F. Lynch Address2 Caldwell St. Death24 Feb 1904Boston, MA Male36 yrs 11 mos 6 daysSingle occupation: Freight Handler birthplace: Charlestown est birth burial place: Malden cemetery: Holy Cross father: Michael Lynchbirthplace: Ireland mother: Mary Harkins birthplace: Ireland
Massachusetts, Births Thomas Lynch Birth Jul 1866 Charlestown, Massachusetts White Male Address 5 Walnut Street registration date: 1866 registration place:Charlestown, MA (p 6 no. 250) fatherMichael LynchbirthplaceIreland motherMarybirthplaceIreland
Kate Lynch Address 5 Walnut St Birth01 Aug 1868 : Charlestown, Middlesex, MA Female White registration date: 1868 registration place:Charlestown, MA (p. 149 no. 387) father: Michael Lynch birthplace: Ireland mother: Marybirthplace: Ireland
Elizabeth Lynchfemale Address 55 Walnut Blvd?? Birth18 Sep 1877Boston, MA registration date: 1877 registration place: Boston, MA (p 3 no. 53) father: Michael Lynchbirthplace: Ireland mother: Marybirthplace: Ireland
Massachusetts, Deaths Mary Lynch Address 2 Caldwell St Death22 May 1907 : Boston,MA Female age66Married birthplace Ireland est birth 1841 burial place:Holy Cross, Malden, MA cemetery: Holy Cross father: Luke Harkinsbirthplace Ireland mother: Mary ConwaybirthplaceIreland spouse: Michael Lynch