兼（か）ねる, a る-Verb meaning ‘to spread across’ (usually 2 different activities or thought processes), often follows the ます-stem of other verbs in order to highlight that ‘(A) is not possible’. However,
兼（か）ねる often indicates that the speaker would actually like to do (A), but cannot due to some external circumstances such as difficulty, already being busy, being hesitant, etc.
This particular grammar pattern will most frequently be seen in its non-kanji form, although both are possible.
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こう言（い）う場合（ばあい）でも返金（へんきん）はできかねます。
Even in situations like this we can’t give you a refund.
それはお客様（きゃくさま）のプライバシーに関（かか）わる情報（じょうほう）なのでお答（こた）えしかねます。
That is information regarding our client’s privacy so we cannot answer that.
お電話（でんわ）での予約（よやく）受付（うけつけ）は受（う）けかねます。
We cannot accept reservations by phone.
兼（か）ねる is somewhat formal, and may be used in cases where a phrase like できない would be considered too strong, so the speaker wants to lightly imply that (A) cannot be done.
Fun-fact - As
兼（か）ねる means ‘to spread across’, this expression can be thought of as similar to ‘to be a stretch’ in English. It just indicates that doing the action of (A) would be a stretch of the speaker’s ability or capacity.
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弊社（へいしゃ）は内装（ないそう）専門（せんもん）なので外装（がいそう）に関（かか）わるお仕事（しごと）はお引（ひ）き受（う）けしかねます。
It would be a bit of a stretch for us to accept exterior work, as we specialize in interiors.