Is it permissible to benefit from the mosque's electricity for a house under construction, with the permission of the mosques' supervisor, knowing that the mosque was built by the father and electricity and water are free, and fearing that the permission might be personal?

If the mosque supervisor is authorized to permit the use of the mosque's electricity, then there is no harm in using it. Otherwise, it is not permissible due to the betrayal of trust involved, as Allah Almighty says: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]." In such a case, he should be advised.