VBA (Microsoft build-in scripting language) will help here. With it, you

• iterate over paragraphs
• check whether one paragraph has "Heading 1" style
• if it does, and if it's empty, delete it

The code would be:
I've tested it with a default template, and it successfully deleted all empty headings, effectively cleaning up the Table of Contents.

sub RemoveEmptyHeadings()
    Dim para As Paragraph
    
    For Each para In ActiveDocument.Paragraphs
        If para.Style = "Heading 1" And _
        para.Range.Characters.Count <= 1 Then
            para.Range.Delete
        End If
    Next para
End Sub

The only counter intuitive code piece is that an empty paragraph is not empty - it has one newline character (causes <= 1 condition).

The quickest way to use this code in a document

• press f11: opens up VBA window.
• locate Project with your document's name and press on it
• 
• closer to the left upper corner, locate an icon between Word symbol and Save button. You can Insert Module by using it.
• 
• Paste this code to the main working area
• Select Run-Run from the ribbon, or press the Run button.

Reuse this code

When you're done with the setup, close the VBA window. The RemoveEmptyHeadings is now accessible as a macro (custom-built little program). One of the ways you can access macros is by pressing alt+f8 and selecting the needed one from the window that appears.

You can also use macros by adding buttons to the Word Quick Access, or assigning keyboard shortcuts. The web describes how to achieve that.