Rehire Maintenance/Processing
There's two ways to avoid double-charging yourself on the Hire Desk rehires, and I have gone into detail below.
Method 1: Order "On Behalf of":
When raising the rehire, enter the hyperlink on the line number on the sales order when assigning the assets to the requisition:

On the "Rehire" tab of the following webpage, check the box labelled "Order on Behalf of".

Order on Behalf toggle:

When this option is enabled, dispatching the items creates the Procurement Plant Purchase Order (PO) only and does not generate an internal dispatch transaction. Financially, this is treated in the same way as not raising a hire through the Hire Desk, as only an external commitment is created. Consequently, the hire is not charged through the Hire Desk, and all associated costs are recorded solely through the Procurement Order and Purchase Ledger invoices.
This hire is then be managed entirely from Procurement | Plant Orders but will still be visible on the operational reporting from the Hire Desk module.

Method 2: "Unlinking" Rates:
Another way we can achieve a similar result (where there is not charges posting from Plant Control, and is all managed by Procurement | Plant Orders and Purchase Ledger) but still retain the hire management functionality from Hire Desk is to unlink the rates when processing the Rehire. This stops the Plant Order element of the process from being locked to the rates entered by the user for the item on the hire desk.
When processing the requisition and dispatch, enter the hyperlink on the Hire line (same as Method 1).
Instead of enabling "Order on Behalf", disable the "Linked" checkbox, and enter the suppliers' price:

Once that is done and the supplier details and cost have been entered, press "Save", and then "Next" to go back to the Requisition lines view. We can see here that the Charge Price is blank, but the supplier price is £568.00p/WK

N.B- If default rates are in use, 'Charge Price' will need to be set to 0.00
With this configuration, any rehire charges posted through Plant Control against the contract will have a value of £0.00. However, when the line is dispatched and the Purchase Order is committed, a Procurement commitment of £568.00 per week will be created.
The second method allows the Hire Desk to continue managing the on-hire and off-hire process for the rehire Purchase Order. The first method removes most hire management activities from the Hire Desk and transfers responsibility for managing the hire period to users responsible for Procurement Plant orders.
If costs have already been posted to the General Ledger twice, they will need to be corrected through a journal adjustment, as there is no supplier account available against which a manual credit can be raised. Alternatively, all related transactions can be cancelled back to the unconfirmed dispatch stage, allowing either of the methods described above to be implemented before processing continues.