In CORE, we have reimbursable expenses (such as client requested extra prints), which should not be confused with the billable expenses . In CORE, we have a 'Reimbursable' check box in the Expense Items and Expense Entries screens that should not be confused with the client reimbursable expenses (billable expenses). This check box is used to track employee reimbursable expenses. They may or may not be billable to the client.
Project-specific, direct expenses should be logged against their respective projects and made billable to the client. The client pays for these expenses.
For your company's in-house and indirect expenses that are not project-specific, CORE can track them as overheads. It is recommended that you create an internal overhead project and track all your overhead expenses against it. Typically, these expenses are not billable nor reimbursable.
There are many reports in CORE that show project-specific expenses and company overhead expenses.