Please help me, re mobile and search position

I hope you are well today and thank you for your questions.

Desktop:
The search icon on my website comes after the last item on the nav. I want it to be in-front of the cart how can I move it to the right. I’ve been trying for ages and used lots of different code.

You can achieve this by adding the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Ivory Search -> Settings -> Settings

@media only screen and (min-width: 1001px){
body #header-outer #top li.is-menu {
    position: absolute;
    right: 40px;
    top: 55px;
    }
}

Mobile:
On mobile I either want the search icon to appear in the secondary nav and work, when you click off it, the search bar goes off the page.

This is the issue of your website mobile menu.

You can resolve the issue by developing a custom code contacting the theme author.

If you are not a developer then you can consider hiring a developer to develop it for you. You can hire a developer from any freelance site or WordPress agency.

You can also consider us to develop it for you using the form https://ivorysearch.com/custom-work/

OR I’ve managed to get the search icon on the main header but it disappears when scrolling and still appears on the off-canvas nav.

To stop this from occurring, you have to add a search icon in your mobile site header by contact the theme support that you are using on your website.

Best regards,
Ivory Search Support

As said above, this is happening because of your website theme functionality so you have to contact your theme author or develop custom code to achieve this.