〜にかまけて
too wrapped up in (and neglecting)
Noun + にかまけて
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にかまけて means being so absorbed in one thing that you neglect something else you should be doing: 仕事にかまけて、家族をほったらかしにしていた, "I was so wrapped up in work that I neglected my family".
The noun is what you're busy with (work, play, childcare, games, busyness itself). The second half is what got neglected, often with おろそかにする ("neglect"), ほったらかす ("leave untouched"), 後回し ("put off") or 〜しなかった.
It often sounds like an admission or a regret. It's also a common excuse: 忙しさにかまけて連絡しなかった, "I was so busy I didn't get in touch".
Don't confuse it with にかこつけて, "using as a pretext": with かこつけて the busyness is fake, but with かまけて it's real.
Examples
仕事にかまけて、家族をほったらかしにしていた。
I was so wrapped up in work that I neglected my family.
遊びにかまけて、勉強をおろそかにした。
I was so busy having fun that I neglected my studies.
子育てにかまけて、自分のことは後回しだ。
I'm so caught up in raising the kids that I put myself last.
忙しさにかまけて、しばらく連絡しなかった。
I was so busy I didn't get in touch for a while.
ゲームにかまけて、宿題を忘れた。
He was so absorbed in his game that he forgot his homework.