この場合、次のような言い方ができます。
ーMy son gets attached to his father.
ーMy son becomes affectionate towards his dad.
「息子が父親に懐く」
「懐く」は to get attached to ... や to become affectionate towards ... を使って表現できます。
例：
When my husband is off work for a long weekend, my son becomes attached to him, but when he doesn't see him for several days during the week, he becomes a bit more distant.
「夫が連休で仕事が休みだと、息子は夫に懐くが、平日しばらく夫を見ないと、ちょっと他人行儀になる」
to become a bit distant で「ちょっと他人行儀になる」と言えます。
ご参考まで！
My son becomes affectionate when we spend time together, especially during extended holidays. But during the weekdays, when I'm at work and we can't see each other as much, he becomes a bit distant and reserved.
こんにちは！
ご質問ありがとうございます。
先のアンカーさんが丁寧に回答されていますので、ここでは他の例を挙げてみます。例えば、
My son becomes affectionate when we spend time together, especially during extended holidays. But during the weekdays, when I'm at work and we can't see each other as much, he becomes a bit distant and reserved.
として、『私の息子は連休などで一緒に過ごすと懐いてくれるのですが、平日私が仕事で会えないと他人行儀になります。』とするのはいかがでしょうか。
役に立ちそうな単語とフレーズ
reserved 打ち解けない、堅苦しい、控えめ、無口な
参考になれば幸いです。
カナダ 
日本 
