Why is Google displaying deleted categories and snippets?

I've been trying to sort this for months. I don't understand why Google is showing subpages, including text underneath, that no longer exists (hasn't done for over a year). For example, I deleted the 'Community projects' category over a year ago, it's not the title of any pages and so I don't get why it's being displayed. Plus the text underneath 'I photograph families in homes...' hasn't been in existence for ages too. 

I've set up a bunch of re-directs but I'm just completely stumped. Can anyone help?

Have you requested that Google remove those pages from their search results? They have a facility to do this.

It is not surprising that G may not have updated their results in their cache for months.

The inverse of asking for a removal would be to ask G to crawl your site again. That is also something G will let you do.

If you haven't set up a Google Search Console account that wouldn't hurt.