When external users(or clients approvers) are in the expense approval path, they may not receive the expense raised by users for approval.
The application may do a "System Approval" on their behalf. When you check the expense sheet generated by the user you may not find the name of the external approver in the list of approvers.
Whenever external users are in a expense approval path make sure users select a project name when raising expenses. The reason is that Web TimeSheet allows setting up external users for projects and not expense types. Ensure the Expense Types are added to the Allowed Expenses group on the project structure so that the expense types are populated when the user selects the project name on the expense sheet. Once the user selects the project name on the expense sheet the external user will be visible on in the approvers section.