A discretionary trust is a flexible type of trust often considered too complicated for most banks liking, the trustees of the trust own the trust’s property on behalf of the beneficiaries. The trustees can pay out income or capital to any one or more of the beneficiaries entirely at their own discretion.
Most high street banks don't like them because they can be confusing or complex and costly to administer due to the individual terms of the trust, which conflicts and does not run smoothly with the banks general practices, rules or policy. Like you state, Barclays seems to be the only high street bank normally willing to accept them.