Is It Possible to Add an Additional Field to My Sign-up Form?

Yes, it's possible. You can edit the sign-up form and either a visible or a hidden field. When creating a form, the second step gives you the option to add additional fields, which are connected to the attributes in your sign-up list.

For each attribute, you can chose if the field should be hidden or if it should be a mandatory field or not. Note, however, that according to GDPR, the only mandatory field you can require from new subscribers, who are not current customers, is their email address.

If you chose to keep the attribute hidden, you can enter a value here. This can be valuable when you want to sort users by how they registered for your newsletter. For example, which language did they use to register? (That can then be the language you send them all communications in.) Or which website did they register from? This is useful if you have several registration forms, as it allows you to see which website the subscriber used to sign up to your newsletter. 

A user cannot see a hidden field in a sign-up form.