SSA always pays one month behind. Your payment for September, for example, was actually for the month of August. Moreover, SSA does not prorate its benefits. There is no prorated benefited owed to the "estate" for the month of September if the decedent was receiving SSA retirement or disability beneifts , assuming that the death occurred in September.
If you are caring for your child (and he or she was the child of the deceased account holder) who is under under the age of 16, you would be eligible for a survivor benefit until the child is 16, unless you earn too much money. (I am assuming that you are under the age of sixty.) If the death occurred in September, then I believe that the SSA benefit payable on your own behalf would be payable for the month of September, but you won't receive it until October (either the second, third or fourth Wednesday of the month), using my example. This payment won't be prorated either.
Benefits on your own behalf (assuming you qualify), generally will end the month immediately before the child reaches age 16. (You likely will receive this payment during the birth month that your child reaches age 16, since SSA always pays a month behind.) You likely will continue to receive benefits on behalf of your child as a representative payee until he or she is 18 years of age. At age 18, the benefits for the child can be continued if he or she is a full-time high school student, but SSA will pay the child directly, baring some special circumstances (child is mentally incompetent becaue of a severe handicap, etc.) These student benefits will be continued until high school graduation or age 19, whichever occurs first. An exception to this one will apply if the child is deemed incapable of self-support because of a severe mental or physical handicap that prevents gainful employment.
I suspect that some of your confusion may be related to the fact that SSA always pays a month behind and because there usually is some lag in the start up of the intial payment which likely was larger than subsequent payments as it likely represented two or three months of benefits.