It looks like you added the logo the the Menu left HTML.
If so, you can put this HTML in a show_if shortcode with the ID of the home page, See the Add-ons page for the details.
It will look something like
[show_if post_id=13] <img src="https://advancedlsperformance.com/wp-content/uploads/2022/07/Advanced-HP-Logo-e1658839778220.png"> [/show_if]
Assuming I am right in the fact the Home page has the page id #13