Is it permissible to supplicate and speak with Allah Almighty during prayer about burdensome matters and wishes with freedom and acknowledgment of sins, or should supplication be limited to simple invocations and speech only?

Supplication is worship, and the best supplications are those found in the Quran and Sunnah. There is no objection to a Muslim supplicating with personal prayers that come easily to them, or with prayers chosen from others, provided they adhere to Sharia guidelines and are befitting of Allah's majesty. It is permissible to express complaints, ask for relief from distress, and generally confess sins. Supplication during prayer is permissible, whether it is from transmitted prayers or not, and whether it is for oneself or for loved ones, as long as it does not involve sin or the severing of kin ties.