Problem with integration fields (Shopify)

I have a problem with my Shopify integration field.

I use prismic to enrich product data from Shopify. I have a model "product" with a Shopify integration field.
Now I have a product object with a product connected, which is deleted in Shopify.
It however is still connected and gives no error. The new product I added in Shopify with exactly the same name is not available in the Shopify product list.
Now I have 2 questions:

What is the process of having products connected which are no longer available in Shopify?

and

Why does a product with the same name not sync with my integration field? the sync gives no errors.

Thanks in advance

Hi Mathijs,

Welcome to the Prismic community.
I will gather some information & do some tests and get back to you,

Fares