to the extent - only as far or as much as
that they relate - that they are related to, concerned with, have to do with
to the cost of activities - the money designated to be spent on this project
which the relevant public authority - the person (civil servant) who was supposed to do it
would not carry out - didn't.
An official earns $100,000 a year. He get's a side-project. It doesn't happen. Someone can put in a claim for his salary, maybe as an expense in a lawsuit or something. But you can only count that portion of his salary that he would have used in doing the job he didn't do. If it ever happened (hence, the "if the project were not undertaken")