On the assumption that the Deputy is referring to a woman who is a citizen of a member state of the European Economic Area, the provisions applying to the children of such a person are to be found in Regulation (EEC) No. 1612/68 of the council of 15 October 1968 on Freedom of Movement for Workers within the Community as amended by regulation 312/76 as well as the European Communities (Aliens) Regulations, 1977 (S.I. No. 109 of 1993) and the European Communities (Rights of Residence for Non-Economically Active Persons) Regulations, 1997 (S.I. No. 57 of 1997), which implement the relevant council directives in this field.
Subject to the above provisions, a residence permit or document is issued. The child of an EEA citizen, who is also an EEA citizen, is normally entitled to work without a work permit as is a child who is not an EEA national but who comes within the relevant definition of "dependant".
If the Deputy has a particular case in mind he might consider contacting my Department to pursue the matter further.