could it help my family court application?

Agreeing to at least some initial supervised contact is very unlikely to make the non-residential parent appear in a negative light.

Arguably, it could be perceived as the non-residential parent doing all that they can to enable them to spend time with their child, and to make every effort to ensure that their child feels safe and comfortable with the contact at all times.

Further, as either of the parties, or the Family Court, are able to request reports of the contact, the reports will be able to be provided to the Court to confirm that the time the non-residential parent spends with the child is progressing well.

In circumstances where there has been a significant period that the non-residential parent has not spent time with the child, or where a parent or child needs to build up the trust for an ongoing relationship, these reports can be useful in progressing to non-supervised contact.