I have just bought my first pair of dedicated cycling shoes, after years of using normal footwear.
The thing is, I bought a pair that I could use with flat pedals or clipless. However, the shoes came without the rubber covers (part of the sole/tread) that cover the screw holes. So i'm worried about using them with flat pedals.
So the question is, do cycling shoes normally come with these covers? And also what is the proper name for the 'covers'?

Covers is as good a name as any. They should be easy to acquire, since anyone who opts to use clipless pedals doesn't need theirs. My bet is that any local bike shop would have tons of them hanging around. Some shoes come with them, some don't.

I have bought 5 pairs of shoes (3 different brands) and never had a pair that came with anything to cover those screw holes.

Most of the many SPD type shoes that I have bought over the years have had some sort of cover

spd or spdsl is a cleat type, not a shoe type.

Often times shoe manufacturers will use the SPD nomenclature to define that it is a cleat compatible shoe vs a non compatible. The word SPD is easier to use than CrankBrothers or another type since it is a short abbreviation.

actually if what you are saying were to be true, it would be false/restricting advertising.... SPD and SPDSL is a shimano specific term... it is short for shimano pedaling dynamics....they invented/released the system in 1990...shoes should be listed by road or mountain compatible, two or three hole (some do both)... road shoes will take the spd-sl as well as other brands of cleats as long as they are 3 hole... mtb shoes same thing only 2 hole.

I hope you call facial tissue just that and not kleenex.

41 Brand Names People Use as Generic Terms | Mental Floss

When someone says "spd shoes" everyone knows what they're talking about, calm down.

I have never seen a shoe add call a shoe an spd shoe, they do say they are spd compatible...

true The S in SPD is Shimano. which limits the terminology application, and so the compatibility thing comes in..

https://www.chromeindustries.com/us/en/footwear/city-spd

Our City SPDs are designed for messengers, commuters and urban cyclists who want the performance of a clip-in SPD in a shoe they can wear all day. Three years ago, we pioneered the City SPD category with our Kursk Pro SPD Sneaker. Our Kursk Pro quickly became a best seller. Our most recent innovation includes our new FlexPlate™ Technology. Truly the most comfortable and walkable SPD shoe on the planet.

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None of any of this nitpicking is helping the the OP to find what he/she is looking for

These aren't the coolest looking shoes, or the lightest, but they are super comfortable, walkable, and easy to drive in, let alone use with platform pedals when needed.

https://smile.amazon.com/Shimano-2014...ano+bike+shoes

The shoes in question by the way, are the Scott Metrix shoe. Don't know if that makes any difference.
I like the shoes, and wanted the choice of using them as flat shoes or with cleats. Suppose I still can.
The shoes don't feel complete in a way though without the covers. Maybe because they are at the lower end (price) of the market.

Again. The OP already has shoes. Just looking for the covers to conceal the gap where the cleats would go
Focus on the original question

If the OP is looking for the covers of a specific shoe, I'd start with that manufacturer and see if they offer covers.

To the OP, the cover is just that, it doesn't offer any support for the shoe, mostly just a preventative for stopping mud and dirt from getting into the cleat. With a cover or without a cover, you should be able to use them with platform pedals...or any other shoe you own could most likely be used with platform pedals.

Those aren't my initials on the toe of my shoe.

My guess is that Shimano is somewhat lax about the trademark, as it all points back to their company. The shoe above, in fact, is a Shimano shoe though. I do admit that I like my Wellgo SPD compatible pedals. Has Shimano's patent protection run out yet?

As far as cleat hole covers, I got my shoes off of E-Bay without them. How secure are the covers?

I'd encourage you to mount and ignore the SPD style cleats. Then ride them with whatever pedals you choose to use. If you have problems with the cleat gumming up with mud, get a screwdriver or something to clean it.

I thought that the covers were only if you were not running cleats. Will they stay on with the cleats on there? The cleats on my Chrome shoes protrude a little as the shoe has a nice curve to the sole. I don't think covers would stay on those.

^ This. I don't think manufacturers intend for anything to go in there. It's just a recess for the SPD cleat.

By the way I think the term is "plate" or at least that's what some manufacturers call them. They made adapter plates or inserts so other brands of cleats can work with the shoes.

A shoe repair shop should be able to make and glue in something. to fill in the sole recess.

some of the 'casual look' shoes were made where you had to remove a piece , molded with the sole, to uncover the cleat hole ...

That isn't accurate. If you use the cover, there is no cleat to protect. What the cover does do is to prevent dirt from getting into the shoe

My Lake shoes came with a cover, but it is slick, because it is a plastic, and not the same rubber as the rest of the sole and I prefer going without it.

The advantage of this style shoe is the cleat doesn't hit the ground or make you walk like a duck. Therefore the rest of the sole should stand up to walking without a cover. I would just use them without a cover (after removing the plate to keep the threads clean for when I did use them to mount cleats).

I spoke to the bike shop I ordered the shoes from. They told me they don't come with covers. Did not sound sure though.
I was expecting something like this as seen on a different Scott shoe . .https://dfp2hfrf3mn0u.cloudfront.net/...tif_zoom_1.jpg
Also the Metrix shoe is described on the Scott website as being 'equally good for cycling and walking'.

MTB shoes generally don't have covers. Some commuting/spinning shoes come with the cleat attachment area covered by a removalble part of the sole that is torn and cut out if attaching cleats. The intent is that it's one way. If you want to put it back, you have to remove the cleats and glue the piece back in. A replaceable one like on that Scott shoe is not the norm in my experience.