If it were possible to separate public services into a public good aspect and a private good aspect, then government could probably ensure better outcomes for the poor by focusing primarily on the public good aspect.
A public good is both non-rival (the consumption of a unit does not reduce the units available for others) and non-excludable (it is not possible to include some while excluding others from this good). For example an illiteracy free community is a pure public good that demonstrates both non excludable and non rival qualities. It is non-excludable as it is not possible to exclude someone from the benefits of an illiteracy free jurisdiction while including others; and non-rival as one person consuming an illiteracy free jurisdiction does not reduce the stores for others. The private good have both rival and excludable characteristics (the consumption of a unit reduces the availability for others and it is possible to include some while excluding others during consumption). Alternatively a school is a private good - it is rival (there are only a certain number of children you can fit in a classroom) and excludable (you can be excluded if you do not meet certain socio-economic standards).
Assuming that all public services have rival and non-rival, excludable and non-excludable characteristics, it should be conceptually possible to separate the public good aspect and the private good aspect.