To remove only specific categories from a list, can use a CSS trick, Attribute

Tutorial: https://css-tricks.com/almanac/selectors/a/attribute/
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Apply this formula to theme search value you want to remove.

Ex: To remove “Sold” from this dropdown

Css Used:

.filter_menu [data-value="sold"] {
display: none!important;
}

And should apply in header search and half map search

Hide specific categories In front end dashboard submit form

Adapt the values for the categories you wish to hide and use the below css in Custom CSS or Customizer

#prop_action_category_submit option[value="179"] { 
display: none; 
}
#prop_category_submit option[value="175"] { 
display: none; 
}

Help article about how to add custom CSS

Tip: If you have theme cache enabled, make sure you clear cache to see instant results.