I'm Miles from the Social Care team at Shopify. Thanks for reaching out about adding a processing fee. Although you're not able to add one in natively, there are a few options for you.
The most simple way to do this is to increase your product or shipping costs? Whilst this doesn't add a processing fee specifically, it does give you a bit more cash for anything you're trying to cover. This avoids your customer seeing extra charges in the checkout which means your less likely to see abandoned carts in the checkout. If you do decide to add it to your shipping rates, we've got a great guide on how to Mark up or discount calculated rates.
If you do wish to display a handling or processing fee, you may need to use an app from the Shopify App Store. One app I would suggest would be Order & Product Fees. Here's a small list of how this app works.
I'd love to understand your reason for adding a processing fee? One of the reasons we don't offer this in the admin is that customer trends show extra charges can deter customers from completing their purchases. This means a higher chance of abandoned checkouts which can affect your bottom line. I'd love to help out with any issues around costing, so please be sure to let me know if you do need a hand at all.