можно

ещё, вероятно, понадобится согласие от второго родителя ребёнка на проживание на Кипре

Family members covered
i. The sponsor’s spouse, provided that the marriage took place at least one (1) year before the submission of the application for family reunification. The spouse must be at least 21 years of age.
ii. The minor children of the sponsor and his/her spouse, including children adopted in accordance with a decision taken by a competent authority in the Republic or with a decision recognizing an adoption of another country or with a decision automatically enforceable under the international obligations of the Republic.
iii. Minor children, including adopted children of the sponsor – as defined paragraph (ii) above, where the sponsor has sole custody and the children are dependent on him/ her.
iv. Minor children, including adopted children of the sponsor’s spouse – as defined paragraph (ii) above, where the spouse has sole custody and the children are dependent on him/ her.
https://www.gov.cy/mip-md/en/documents/family-members-of-certain-third-country-nationals-that-are-legally-residing-in-cypurs-fr-family-reunification/

я не совсем прав, нужна единоличная опека
"where the spouse has sole custody and the children are dependent on him/ her."

в списке документов - выписка из соц. инса. вашей работы
если она у вас есть - можете податься сразу на рабочую супруги киприота