I have enough private money that I can take time off of work to have a meeting with the politician for my district. I might, for example, advocate that government computers use standard document formats rather than proprietary ones. Or I and co-workers may chip in to send one of us to petition OSHA in person, to increase workplace safety requirements.
The National Woman's Party is an example of a lobbying group which worked for decades first for women's suffrage and then for prohibition on sex discrimination and the Equal Rights Amendment. Where was the corruption in that use of money to influence politicians?
In your example of a poor single mother working two jobs, that mother is very unlikely to have the private money to take the time off. Thus, private money will always have an effect on politics.
There are ways to limit its effect. For one, tax the richest people and most profitable companies much more heavily than they are now, and increase social support for the poorest. But by your criteria that's still going to end up with corruption.
The people should be able to influence a politician more often than just at the ballot box. The question is, at what point is that extra influence inappropriate? I believe your use here "corruption" is too strict and binary, such that no government can meet that standard.