Inappropriate training activities
These sort of activities can leave you personally liable for claims and the club will not be able to help you.
Take a look at this from the Kent FA for more details and a story about an unfortunate coach here
Play an "unsanctioned" match or training game
Games are sanctioned when the County FA has approved them. The "approval" means that the County FA's public liability insurance applies to the game as detailed in the approval request.
Without public liability insurance you as the organiser of the game assume liability and become responsible for any costs associated with damage to third party property. You are also liable for any costs associated with any injury (and long term care etc.) to any third party.
Imagine if one of your players miss-hits a ball and it strikes an elderly person in the head who happens to be out for a walk. Whatever injury is sustained, you will be liable for the costs of defending and perhaps if you lose the medical care costs, if they decide to pursue a claim against you.
Arrange a training session without club approval
This falls into the same category as an unsanctioned game (see above). Any training session you arrange without club approval will not be covered by public liability insurance.
These sessions often cause other problems for the club. Many local pitches/grass areas will require booking and approval by the owners before use. We often have to field complaints as a result.