Insurance has its origins in self-help groups which came together to provide mutual protection against a peril faced in common by all the members of the group. These mutual assurance societies are now less usual than private companies who undertake, in return for payment of a premium to pay out in respect of your losses caused by various specified events.
In the case of public liability insurance, those events will tend to be claims made against you in respect of alleged negligence of various sorts causing loss to a third party. It is common for some types of claim against you to be outside the scope of this cover. For example it would be unusual to be covered under a policy of this sort for a claim in a tort such as false imprisonment or defamation, perhaps because these carry an element of intentionality.
Where the limits on public liability insurance are too tight financially or in the scope of the cover, it is possible to take out an umbrella policy which not only provides for payments above and beyond the amount of cover under the chief policy, but can also be designed to cover potential insurance claims which are completely outside the scope of the more regular sort of policy. This could include the torts mentioned above in some situations, but will probably not extend to cases where you are accused of deliberate wrong-doing.
It is unlikely that you will be covered under either a regular policy or an umbrella one, for acts which involve a large element of intention on your part. Cover of this sort would be against public policy, so even so-called umbrella policies will not embrace a claim of that sort. The reason is that if a person planning for example a criminal assault was able to protect himself against the financial consequences, then that could be seen as enabling and promoting what is actually a crime.