If a customer has out-of-stock items in their cart and you have selected Allow customers to purchase this product when it's out of stock on the listing itself, the customer can choose to signify whether they want their order shipped only when the inventory is all in-stock, or to ship the inventory as it’s made available. To turn this setting on, you'll just need to add a tick to the 'Enable Ship Partial Option' box within the 'Settings' section of our app.
Once this is turned on, this message appears in the cart if there are backordered items within the order, and will be shown as below to your customer:

Please note that this will default to 'Ship Partial', so this will be listed as their choice if they don't choose otherwise.
Their shipping preference will be added to the notes section of their order for your reference:

Please note: This setting is simply a way to allow them to signify their preference; whether you ship the order partial or complete is entirely up to you at the end of the day, and this is just a tool to help communicate their preference to you without further correspondence being required.