How to approach late payment situation, especially with a PM?

[Turgut Oz]

I have a rental property in Texas and I hired a local management company to manage it. They placed a tenant who did not pay the first month on time. The rent is due 1st of the month. The tenant paid two partial payments for that month later and still owes an unpaid portion from that month. In the second month, I noticed he did not pay the rent again and PM says (upon my asking about the situation) the tenant contacted them and told that he will pay half in 10th (half of second month's rent), and rest about 17th. So I am facing a situation where I definitely do not get payments on time, and also not in full.  Especially, it is not just late payment, it is partial late payments scattered over the month.

My property manager takes 100% of the late fees so it is actually better for them if a tenant pays the rent late every month. They did not even bother to let me know these partial payments plans if I did not contact them. While I know this is only a two-month experience but I also know that these things become habitual, occurring every month if the tenant knows it is OK to do that. Having payment issues in the very first months is a strong indicator that they placed a wrong tenant. 

How should I approach this property management? What are reasonable things to expect and demand from them? Currently, I feel like they do not care. Also, how many late payment patterns form a basis for eviction if this thing continues in the future?