1. The corresponding option may be disabled in the extension configurations.
Please proceed to Stores → Configuration → Amasty Extensions → Free Gift and check the Add Simple/Virtual/Downloadable/Bundle Products Automatically, Without Confirmation option. In case it's set to No, promo items won't be auto-added.
2. Promo item is not a Free Gift but a discounted item.
For discounted items to be auto-added to the shopping cart, please make sure that the Add Simple/Virtual/Downloadable/Bundle Products Automatically, Without Confirmation is set to Yes, discounted and free products.
3. A promo item is a configurable product.
The Free Gift extension does not add configurable products to the shopping cart automatically since customers should select preferable options before they can be added.
In case you intend to add a certain option of the configurable product as a Free Gift for customers, you will need to specify the SKU of the child of a configurable product. For example, let's say you want customers to receive a purple XL-size tee as a promo item which is a simple product and has its own SKU - WS12-XL-Purple. Please insert this SKU in the Promo Items field instead of the configurable product's SKU:
Once done, the child product will be auto-added to the shopping cart: