If we have a cosmic microwave background should't we also have a cosmic radio wave background? I'm a layman in physics, but here is what I understand:

*

*What we see in the sky with naked eyes is a map of electromagnetic waves in the frequency visible to the human vision.

*But that kind of observation is too limited because as the universe expands, the light coming to us redshift to the invisible spectrum. So we started observing the universe in lower and lower frequencies.

*The James Web telescope for example, can see further distances because it has infra-red light sensors.

*If take this idea to the extreme and detect waves from the microwave spectrum, mapping the entire sky with this method, we get the CMB.

*By seeing further objects, we also see these objects in the past, taking account the time the light took to reach us (and the expantion of the universe).

*By the model we have today, the CMB would be the furthest thing we could observe, because after that, the universe was opaque.

So what I am trying to ask is:
Does anyone tried to listen beyond microwave and create a map of background radio waves?
I know many objects can emit radio waves. So we probably would detect these objects and also interference from manmade radio waves from earth.
If there is no background radio wave, the image would be probably one color with the interferences I cited above.
But how can we know we wouldn't find anything if we try to detect longer and longer waves?
How map of radio waves in the sky would look like?
 A: High energies imply short wavelengths and long wavelengths imply low energies. The original wavelength of the energy release from electron capture during recombination is more or less the average ionization energy for the same atom (H or He) which has a low-frequency cutoff below the ionization energy. This argument suggests that at recombination, there wasn't much longer-wavelength radiation zipping around which could subsequently be redshifted into radio frequencies below that of microwaves.
You can see this in the CMB spectrum shape, which is a nearly-perfect blackbody spectrum that has cutoffs above and below the peak at 2.7 degrees K.
There is a radio wave bath in space at lower frequencies but it represents the noise floor created by RF radiation from things like active galactic nuclei and not from the CMB.
A: In the future we can use longer wavelengths to look further into the early universe, but not further back in time.
The problem with using longer wavelengths to look further back in time than the CMB is that the universe was opaque at that time.  For the first roughtly 370,000 years the universe was opaque.   Photons would not travel far before being absorbed.  At about the 370,000 year mark, the whole universe cooled down enough to become transparent.   The universe would have been awash with photons colliding with the plasma and being re-emitted at the temperature.   As the 370,000 year mark neared the mean distance between being absorbed and re-emitted was only on the order of 100 light years.  Very quickly this changed to be the size of the observable universe making the universe transparent.  The problem with seeing further back in time is that something emitted at the 369,900 year mark would have likely been absorbed within 100 years.
The expansion of the universe has caused those wavelengths to increase so that radiation now peaks at around 900 micrometers, and we are seeing further into the early universe.   In another 13 billion years if everything goes how we think it will, the cosmic background radiation from the 370,000 year mark will have been redshifted even more into the radio wavelengths and we will see further, but only to the same time.
Imagine a straight line pointing from the Earth (A) out into the universe to
other points in the early universe that existed 370,000 years after the big bang along with the light created by that point at the 370,000 year mark as a lower-case letter (i.e. b).  Here we see the light is at the point where it was emitted.  Any light emitted before this from point B would have been absorbed and re-emitted (scattered) before it got to A.
abcdefg
ABCDEFG

Imagine a dense fog where you can only see a few feet in front of your face.  the light 'g' at G would get absorbed before F and re-emitted, so there is no way to see further into the past than this time. As the universe dropped below 3000K, the universe quickly became transparent and the CMB we see today is from that time.
As the universe expanded, we could see further and further into the universe, but only back to the time when recombination occurred.  Imagine enough time has passed that we are seeing light from C when recombination occurred and the universe has been expanding.   The photons from C-G have been making their way towards A.  We now can see what has been going on at B since the last scattering and can see the light from C at the last scattering.  We also see that B is moving away from us.  However, the light from F and G at the last scattering is actually further away from us than it was at the time because the universe is expanding too fast.   We will never be able to see that light.
ABCDEFG         (original distance)
c-d-e-f-g       (light from last scattering)
A-B-C-D-E-F-G   (current distance)

If we continue further into the future:
ABCDEFG               (original distance)
d--e--f--g            (light from last scattering)
A--B--C--D--E--F--G   (current distance)

Notice the light 'f' is not getting any closer to us.  This is the cosmic horizon.  Assuming this is the current time, the light 'd' has been travelling towards us for 13.8 billion years and has redshifted from infrared into microwaves.   We will eventually be able to see 'e' when it has redshifted further into radio waves.  However this is not seeing further into the past, the light 'e' was emitted 400,000 years after the big bang at the same time as 'd', just in a different location.
A: Yes, there's a cosmic radio background, just like there's a cosmic optical background, infrared, x-ray, etc. The background from AGN and whatnot that @nielsnielsen mentions doesn't kick in until about 300 MHz, or so, so there's definitely a good chunk of radio wavelengths where the background is dominated by the signal blackbody signal from recombination.
See, especially, figures 1 and 8 from the review by Hill et al (2018) published in Applied Spectroscopy (also from arXiv).
The trouble you're going to run into when trying to look at longer wavelengths of light to see farther back in time is that the signal from the CMB is from when the universe changed phase from a plasma to being neutral in a process we call "recombination." The temperature of the universe was around 4,000 Kelvon then, about like a red dwarf star. Plasmas have an interesting property called the plasma frequency. Basically, plasmas are opaque to any light with a lower frequency than that. For higher frequencies you have to look at something called the mean free path of the photon.
My understanding is that, for all intents and purposes, the universe was opaque before recombination. So it is highly unlikely that any light from before then was able to travel in a straight line for long enough to give us any information about where it came from before it bounced around in the universal plasma.
That's part of why there's interest in finding a signal from neutrinos or gravitational waves, since they'd be able to travel through the plasma without any trouble.
And, yes, we have maps of the sky at radio wavelengths. I don't know if they're sensitive enough to look for structure in the CRB (cosmic radio background). One challenge is that most radio observations are done with interferometers, and they reconstruct their images in a way that removes large scale signals. You're really best off with single dish radio surveys, like could be done with Arecibo, and can be done with FAST. See, for example, the maps created by GALFA. Their interest was local HI (neutral atomic hydrogen), not CRB, so I don't know if their data is sensitive enough to detect any cosmic signals.
