You can cure the mortgage by paying all amounts due, including interest and late fees. The bank will refuse to take partial payments when you are behind, but they will accept full payment to bring the mortgage current.
The only reason mortgage companies refuse payment is because they have not received payment in full of all past due amounts. By refusing payment, they are preserving their right to bring a foreclosure against you.
If you have the funds to bring the mortgage payments current, put them on the table. Otherwise, the mortgage company is saying that the past due amounts will be increase by foreclosure fees, late fees, etc when they take the next step.
Hope this perspective helps!