Highfields Community Association (HCA) runs the Highfields Centre, in Melbourne Road, and is at odds with Sir Peter over funding.
The HCA operates independently but had received £293,000 annually from the council as it sought to become financially self-sustaining.
Sir Peter said the HCA failed to meet that target within the agreed three years and the council could not afford to maintain the level of subsidy.
The HCA has accused the mayor of acting autocratically and being dismissive of the community in Highfields which uses the centre.
Sir Peter in turn said the leadership of the HCA had been acting irresponsibly.
The mayor recently wrote to HCA chairman Furzana Khalifa expressing concern that centre staff had been withdrawn from the local government pension scheme without a new one apparently being set up to replace it.
HCA head of centre Priya Thamotheram said: "It is deeply worrying that a centre of excellence can be subjected to the city mayor's autocratic whims. We hope exposure of his practice to a wider audience will lead to a more reasoned outcome."