Pain and suffering damages are an element of damages for a tort suit, not a landlord tenant lease.
In most tenant arrangements taking out the garbage on a daily basis falls to the tenant.
Dear the property owner said the house will be in foreclosure:
New York State does not allow for a separate lawsuit for pain and suffering. A tenant may sue a landlord for a personal injury arising from illegal conditions in the home, or arising from violations of law. If you are interested in pursuing legal claims, make certain that the conditions of the home are inspected and recorded as violations. To pursue a claim of injury to the children, their doctor would need to report that their exposure to disease occurred in the home. You may not want a record like that to come into existence.
In any case where a party is injured by the negligence of another party, in New York State, a jury is allowed to consider whether the injured party in some degree contributed to their injury by their own negligence.
If you associate your children's illness to the unhealthy condition of the garbage on the porch, don't wait for the landlord to clean up the mess.
You could withhold rent for the dangerous conditions, but you may end up being sued for non payment of rent. In a landlord and tenant lawsuit, a tenant may not ordinarily raise a counterclaim for personal injury. If a tenant does join a personal injury counterclaim, the court likely would sever the counterclaim. In any case, you do not want to bring a lawsuit for a personal injury for injuries to your children without an attorney.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.