Instead of going straight to seeking legal advice it might be worth asking CSA to review the assessment first. Then you know you have an up to date figure to work from with negotiating (or is the current assessment fairly current and based on current information?).

But there is no specified guidelines on what child support should 'cover'. It is to help with the expense of raising a child, whatever expenses it can based on parental income and care percentages. It is a conservative figure and rarely will be able to contribute substantially to the costs that raising a child actually entails but it is, at the end of the day, all a person is legally required to contribute so if that is all your partner feels he can contribute, he could scale back at any point IF this extra expense is not a formal agreement that would need to be challenged first.

But please, please do tread gently. If this arrangement has been in place for a while (before you became involved especially) it will be hard to swallow that things might need to change.