What is the ruling on reciting the verse: "O Lord, even if the supplications come to You broken, spread the wing of contentment and mend our hearts"?

We do not find anything in these words that prevents one from supplicating with them. It is permissible for a Muslim to choose the supplication most appealing to him, even if its wording is not established in the Sunnah, due to the general saying of the Prophet, peace and blessings be upon him: "Let one of you choose the supplication most appealing to him, and then let him supplicate to Allah, the Mighty and Majestic." Based on this, there is no harm in supplicating with anything whose meaning is sound and does not contain any Sharia violation.