It's a honeycomb - in effect a thick piece of metal with a lot of small holes drilled through it (although not made that way). To save time, I've just cut and pasted the section about them in my "Lighting Magic" book - which you have and which might be useful
"Honeycombs
Standard Reflectors are often used with a honeycomb (known in the USA as a grid) which concentrates the light and which is extremely useful.
The better standard reflectors have a special lip machined into them, which holds the honeycomb(s) in place. Standard reflectors are also used with umbrellas.
Honeycombs are generally available in 10, 20, 30 and 40-degree variants.
What a honeycomb does is to restrict the spread of the light, forcing it into straight lines and preventing it from spreading into areas where we don’t want it to go.
We generally use them to restrict the light to a small area, for example to light the hair of our subject, without the light also going on to the face, clothing etc.
We also use them to skim light across the subject, to create harsh local shadows. This is often used to emphasise and reveal texture, and we use them as a backlight, i.e. pointing from behind the subject towards the camera lens.
When a light points towards the camera lens it usually creates unwanted flare, but a honeycomb will stop this from happening as long as the light is pointing at an angle greater than that of the honeycomb. In other words, if the angle is more than 10 degrees then there will be no visible flare if you use a 10-degree honeycomb. We sometimes want to create flare in our subject and the honeycomb is again an essential tool, because it allows us to create exactly the amount and level of flare required, i.e. it gives us control of the flare.
The most useful Honeycombs, for most people, are the 10 and 20-degree versions. Sometimes though, and especially when using very precise lighting, it’s good to be able to use a 5-degree honeycomb, or even less.
The difficulty with this is that, as far as I know, they are made by only one firm, in the USA, they are very hard to obtain and they cost a fortune.
Because of this, when I worked with Lencarta I arranged for special honeycomb reflectors to be made that will accommodate two honeycombs together. Fitting two honeycombs at the same time, and rotating them as required, allows extremely tight and precise honeycomb effects to be created."
And now I'll add a bit more detail. . . We use them in three very separate ways.
1. To put light into a very specific area, that doesn't go where it isn't wanted. In this shot, I used one to light the inside of the reflector, which otherwise would have been dark.
View attachment 313230
2. They're also used to skim along a textured surface, to emphasise the texture by creating strong local shadows.
3. They're also used for backlighting, where the light is behind the subject and pointing more or less towards the camera lens. With a 10-degree honeycomb, as long as it's pointing more than 10-degrees off axis it can't create lens flare.
And of course, they have other uses too, for example as a hairlight. They're mentioned 140 times in my book . . .
I'm not laughing really - I'd like one too.... and a lot else besides! 
Chief