What is the ruling on acts of worship performed with the intention of rectifying one's children, such as giving charity when they falter or performing Qiyam al-Layl with Surat Al-Baqarah and supplicating with the intention of rectifying them? Are there any prohibitions in this? And what is the correct action obligatory upon a Muslim?

It is permissible for a believer to perform righteous deeds seeking Allah's countenance while also desiring worldly rewards, and this is a permissible form of seeking intercession (tawassul). For the texts mention worldly rewards for righteous deeds. And a righteous deed may be rewarded to the believer through their offspring.

The most complete approach is for the believer to intend only the reward of the Hereafter, because the blessings of this world diminish the reward in the Hereafter, even if one did not intend it with their deed. For spoils in warfare diminish the reward, and whoever dies without having consumed any of their reward is better off. A little favor in this world diminishes from much in the Hereafter.