Trust funders will not automatically fill the gap left if your local authority has withdrawn the annual funding it used to give you due to its own funding cuts.
If you used to receive £30K per annum from your local authority, why should a trust fund step in and simply replace that money?
However, do not despair; all is not lost. What the trust funders mean is that they will not automatically plug the gap. What they will do, however, is to consider a fully thought out and well-presented proposal in the same way that they will consider any other proposal. This is not a bad thing. It means that your proposal will be subject to the same rigorous selection process and accountability measures as every other one. This can be a good opportunity to review the work you are doing and be clear about what you are doing, why and how you are doing it.