If a man has 3 children from different women, can he go abroad?

Yes, it can. Provided that he is not deprived of parental rights, pays alimony or supports (financially) children, children under the age of 18, etc.

According to Article 23 of the Law "On Mobilization Training and Mobilization", conscripted men who have three or more children under the age of 18 who are dependent are not subject to military service during mobilization.