買うつもりはなかったが、子どもにせがまれて、つい買ってしまった。
• せかされて
• もてなされて
• 嘆かれて
• ねだられて

The answer is ねだられて.

"I had no intention of buying it, but my kid begged and pleaded, so I ended up buying it."

All of the answers are the passive inflections of their respective verbs, so this means the speaker (presumably the parent) was receiving an action of some sort from a child.

If the parent wasn't planning on buying an item but gave in due to something their child did, it follows that the child was requesting the item.

せがむ, from the question, means "to pester/beg".

急(せ)かす -"to rush/urge".

持(も)てなす -"to entertain".

嘆(なげ)く -"to grieve/regret".

強請(ねだ)る -"to pester/plead/coax".

From these answers, either せかす or ねだる could make sense. However, given that there is no time mentioned in the question, "being rushed" would not necessarily result in the parent caving to their child.

Therefore, the most logical answer is ねだられて (from ねだる).