Google uses neither meta descriptions, nor H1 pages titles for snippets that appear in Google search results. It takes text fragments that are irrelevant for our target audience. Such text snippets don't reflect the basic idea of the page. Consequently the CTR drops down.
Is there any way to make Google use the meta description (or a specific text from the page) to form snippets?
There are no such tools, but the following Google info from the guide Create good titles and snippets in Search Results may help:
Google's generation of page titles and descriptions (or "snippets") is completely automated and takes into account both the content of a page as well as references to it that appear on the web. The goal of the snippet and title is to best represent and describe each result and explain how it relates to the user's query. We use a number of different sources for this information, including descriptive information in the title and meta tags for each page. We may also use publicly available information, or create rich results based on markup on the page.
and
You can help improve the quality of the title and snippet displayed for your pages by following the general guidelines below.
also:
How snippets are created
Snippets are automatically created from page content. Snippets are designed to emphasize and preview the page content that best relates to a user's specific search: this means that a page might show different snippets for different searches.
Site owners have two main ways to suggest content for the snippets that we create: rich results and meta description tags.
Rich results: Add structured data to your site to help Google understand the page: for example, a review, recipe, business, or event. Learn more about how rich results can improve your site's listing in Search results.
Meta description tags: Google sometimes uses tag content to generate snippets, if we think they give users a more accurate description than can be taken directly from the page content.
Your message about the fact that Google is ignoring your H1 content may be a signal about:
Check out the Google quote:
Use heading tags to emphasize important text Use meaningful headings to indicate important topics, and help create a hierarchical structure for your content, making it easier for users to navigate through your document.
Best Practices Imagine you're writing an outline
Similar to writing an outline for a large paper, put some thought into what the main points and sub-points of the content on the page will be and decide where to use heading tags appropriately.
You can check the validity of HTML and content hierarchies using online validators:
Select the Show Outline option on these validators which will show you the existing hierarchical structure of the web page. Alternative: use the online HTML 5 Outliner.
