I think I may have blown a power valve when I first fired the engine up and it backfired through the carbs. Having issues on low and mid range acceleration when I put my foot in it. As near as I can figure out the partts book calls for a 32 power valve which they call a power jet. Will the power valves from a later flat bowel Holley fit and work? Is there a source for individual power valves? Suggestions on what # valve to use other than the 32?

The number on the power jet is the size in thousandts of the fuel feed hole on that particular valve.  Varying that number up or down will richen or lean the fuel mixture accordingly only during moderate or full throttle opening.  At idle and cruise, the power jet is expected to be closed.  The power jets from other Teapot carbs will work but the sizing must be watched for.  I don’t see an issue with larger valves but would be cautious with power jets with smaller than #32 numbers and this is assuming you are dealing with factory dual quad carbs and not retrofitted single four barrel teapots.
 
The brass power jet is typically unaffected by carburetor back fires but the rubber valve assembly itself that attaches to the bottom of the fuel bowl cover could potentially lose its seal to the fuel bowl cover and have to be readdressed.  The rubber valve assemblies are touchy to get a good seal when all is right; having a good seal on that part is easily checked during carburetor assembly.

Lorena, Texas (South of Waco)

Can I use the late model flat top power valves that are available or do I need Tpot specific ones? I am not good with carbs and not familiar with the rubber valve assembly that you refer to. I am aware that a blown power valve on a flat top is a problem and I know for sure that they are totally different carbs but that there are some parts like secondary diaphragm springs that interchange. Still learning about carbs. Could the backfire have caused a problem with the rubber valve assembly? The carbs were professionally restored and have not been run before. It starts, idles, and cruises fine. Ease into it and it's fine. The problem is when I put my foot in it.

You may be confusing the spark advance valve on the side of the Teapot carburetor with the power valve.  The factory equipped 1957 dual quad Teapots do not have the spark advance valve but the single four barrel equipped Teapot engines did have the spark advance valve.  The spark advance valve on the single four barrel equipped Teapots is located on the right side of the carburetor.
 
The power valve or enrichment circuit on the Teapots is a two piece affair with the brass power jet located between the jets and the actuator (valve assy) attached to the fuel bowl lid with the two secondary jets and a single screw.   On the following illustration, #9594 is the power jet while #9A565 is the valve assembly.  The spark advance valve is not evident in this particular picture as its on the lower body of the carburetor.
 

Lorena, Texas (South of Waco)

Here’s the exploded view showing the spark valve or spark advance valve on the right side of the carb.
 

Lorena, Texas (South of Waco)

Being a Teapot 4V carb, the O rings located at either end of the brass secondary tubes may not be sealing thus not having enough negative pressure signal on those tubes for fuel to feed into the secondary throats when the secondaries blades are open.  A quick (and temporary) fix is to put some white grease around those O rings to help seal them up.  You might want to verify that the secondaries are even attempting to open as the gaskets for the secondary housing can be installed incorrectly thus preventing the secondary side of the carb to not open.
 
If those are not factory dual quad carbs, then the fuel metering will not be exactly like the dual quad carbs as the single 4V carbs have more air flow on the primary side than the factory dual quad carbs.  The factory dual quad carbs also had a shared rubber line between the two carbs to allow both sets of secondaries to work in unison.
 
The question at this point is if you have the factory dual quad carbs or retrofitted single four barrel carbs.  Tuning aspects are different between them.

Lorena, Texas (South of Waco)

I have an e setup on my engine that had bad power valve diaphragms on the teapots. If the diaphragm is bad the power valve will stay open all the time including idling. Your fuel mixture will be rich across the board. You could smell the over rich mixure at Idle and the plugs were getting loaded bad. I suggest pulling your spark plugs and see if they are getting sooted up. If they are white or light tan your power valve is probably ok. Because the vacuum passage to the power valve diaphragm leads to the intake manifold I would say a backfire could possibly damage the diaphragm. As previously stated it is easy to install the diaphragm incorrectly and cause a problem. Again get a plug reading and see if you are running too rich.
 

I was confusing the spark advance valve. Carbs are original E dual quad carbs as I stated earlier. They worked fine before restoration but were always a little leaky. Winter here in Illinois so not going to test drive it. Trying to sort out if I need parts while I have time. I need to go through the basics again like valve adjustment and timing and check float levels etc.

Original ‘E’ code carburetors do not have spark advance valves.  You can verify your carbs by looking at the List numbers which are located on the back flanges.  The ‘E’ code carb List numbers are 1434 and/or 1437.  If you have List numbers other than these, simply post them and someone will give the exact original application for them.

Lorena, Texas (South of Waco)

They are E carbs. I was looking at the parts book not the carbs. The parts book showed the spark advance valve on the E carb. One ;picture for E and '56 single carb.