How can I add funding information to a deposited record to ensure it is linked to a specific funding source?

You should follow the OpenAIRE Guidelines requirements to expose the records' metadata.
For each version of the guidelines, we provide specific information on how to populate the funding metadata. e.g. for the OpenAIRE Guidelines for Literature Repository Managers v4: https://openaire-guidelines-for-literature-repository-managers.readthedocs.io/en/v4.0.0/field_projectid.html#aire-fundingreference

You can populate the funding information not only for the European Commission's but also for all the other Funders.
In OpenAIRE, to correctly identify information at the grant level, we also need to collect data from funders.
Please note that in OpenAIRE, to correctly identify information at the Grant level, we also need to have information collected from Funders. Otherwise, there will be a connection to the Funding programme, but not to the Grants.