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You are entitled to support from the father even if he leaves the country. You will want to get in touch with a family lawyer asap to get the paperwork started for the courts. The courts in Ireland can enforce a order even in Slovakia the only issue is you must have a order first.
You are entitled to at least child support and possible maintenance from him as well if you were married. My recommendation is to speak with a lawyer in Ireland to protect your rights and get a order in asap.
You are fine, no need to apologize. His employers do not have to give this information to you or your lawyer. You lawyer can help you locate his home address for you though running his information and searching records. At the least if your lawyer can locate his employer he can petition the courts to require the employer to disclose his legal address.