As per my two previous replies to very similar questions regarding trying to use the LG ultra-wide screen via HDMI.
Some Mac models only support a maximum resolution of 1920x1200 via their built-in HDMI ports. See Frequently asked questions about using HDMI with Mac computers - Apple Support
The MacBook Air which I am presuming you are enquiring about does not have a built-in HDMI port. You are therefore correct in trying to use a Mini Displayport to HDMI adapter.
Not all Mini Displayport to HDMI adapters are created equal. Some in fact many of them are cheap passive adapters, depending on the capabilities of the Mac they can only do at best 4K at 30fps, they all should be able to do 1920x1080 at 60fps. Your ultra-wide screen falls somewhere in between these two levels. The more expensive active adapters can do 4K at 60fps, obviously 4K is more than enough to do the resolution of your ultra-wide screen.
I therefore suggest you get and try an active adapter. This did completely cure the problem for one of the other two people with a similar query although that person has a Mac Pro 2013 computer. The other person has a Mac mini 2014 and has not yet reported the results.
The adapter I strongly recommend is this one - http://www.club-3d.com/index.php/products/reader.en/product/mini-displayport-12- to-hdmi-20-uhd-active-adapter.html