Ah ok so all I had to do was add this to the end of my web map URL: &mobileBreakPoint=100
'100' being the amount of pixels the map width has to be before it switches to mobile mode with mobile pop ups. I believe the default is 600. So change it to whatever you want.
So my final embedded code is:
<style>.embed-container {position: relative; padding-bottom: 80%; height: 0; max-width: 100%;} .embed-container iframe, .embed-container object, .embed-container iframe{position: absolute; top: 0; left: 0; width: 100%; height: 100%;} small{position: absolute; z-index: 40; bottom: 0; margin-bottom: -15px;}</style><div class="embed-container"><iframe width="500" height="400" frameborder="0" scrolling="no" marginheight="0" marginwidth="0" title="Sinkhole_map" src="FULL URL OF MY WEB APP MAP&mobileBreakPoint=100"></iframe></div>
Now I'm getting the normal 'desktop' pop-ups on my embedded map.