I have been using a B&S air filters in my Honda's for years. The part number is 491588s. I buy them at HD for $4.99 and they fit perfectly. When I needed to replace my Honda's AF many years ago, th B&S filter is what was given to me at a local shop. I have been buying them ever since at HD.
I take care of mowers for a bible camp near here. Had the same problem missing air filter cover on Honda mower like yours.
went to Honda dealer new one on hand and for $2.39 if I recall
cheap enough to have a spare.