[Debezo]

～ものか/もんか is a 反語 expression.
it can be a strong negation if it's sticking on the end of the affirmative phrase and with strong tone.
だれがそんな事をするものか. How could anyone do a thing like that?
俺が行くって? 行くものか. He says I'll go? Like hell I will!

however, if it's sticking on the negative phrase, it mostly means "I'm troubled about...", "how should I do...".
and -monokane is used in a roundabout way, "Can't you...?"
もっと早くできないものかね Can't you do it more quickly?

so, that may be
それにしても文章のチェック。どうにか見てもらえない ものかね。
but then, as for the sentences, can't you have him/her check it somehow?