Our first mh had a cover like Dutch's. Our 2nd didn't so we used a baby gate between the counter and barrel chair (which locks into place so the baby gate stays tight).
The other thing we did was use a buggy cord to keep the screen door from opening if it was pushed against (the regular door won't open unless you use the handle but the screen door will open with enough pressure). Basically just snaked the bungy cord from the handrail on the steps, through the screen door hole to reach handle and then wrapped it around the grab handle outside.