It just so happens to be that we say a longer Tachanun on Mondays and Thursdays. It also just so happens to be that we read the Torah publicly as part of the service on Mondays and Thursdays. Is there any connection between the two?
Related: Tachanun on Monday and Thursday
Jewish Prayers: Tahanun states
The reason that extra sections are added on Mondays and Thursdays is because, according to tradition, these days are favorable times for God to respond to our pleas. Ever since court sessions began to be held on these days in the Temple era, they have been seen as days of judgment. Also, Moses ascended Mt. Sinai on the fifth day of the week to receive the second set of the Ten Commandments, and he descended on the second day of the week.
We see from the Torah and from Ezra that one must not go more than three days without learning Torah. See Why is the Torah read on Mondays and Thursdays? Additionally, since people came totown on those days, these became the main market days as well as the days that the courts sat.
As we see, the reasons for the two are similar, though independent. Thus, the resons for both led to them being done on the same days.
Another reason that I have heard was that since Monday and Thursday were longer davening in any event because of the torah reading, the longer tachanun was established for those days.
