Can a landlord give you a 3-day to pay or vacate and then in return accept a money payment towards money owed?

Does a rental lease aggrement get broken if the landlord accepts some money after being given a 3-day to pay or vacate?

First its not a good idea to piggyback your question on someone else's. This can lead to confusion. You should start a new thread. So I've moved your question to its own thread.

Yes and no!

If you are given a 3 day pay or quit notice, the landlord can accept a partial payment and still proceed with the eviction. Only a FULL payment of the arrears can cancel the eviction.

Not paying your rent on time puts the tenant in breach of contract. The landlord is entitled to collect what is owed. There is no way they breach the lease by accepting a partial payment.