In Programmed Mode "P", the camera sets shutter speed and aperture for optimal exposure. Recommended for snapshots and in other situations in which there is little time to adjust camera settings. If one uses flash in this mode and pictures comes out underexposed, would exposure compensation correct this? Please advise?

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Shooter
Nikon D80 DSLR Camera
AF-S DX Zoom-Nikkor IF ED 18-135 mm f/3.5-f/5.6 G Lens
Nikon Autofocus Speedlight SB-800

Hello,
have you tried increasing the exposure setting to compensate for the dark photo ?.
Also please provide more information on the surronding lighting and upload the picture if you can.
Knowing the element will be better to advice and might even hold the clue at to why the image is underexposed.

No, I haven't tried increasing the exposure compensation but if I did that would the picture be properly exposed or I need more light?

Here's a sample of what I am talking about in this photo.

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Shooter
Nikon D80 DSLR Camera
AF-S DX Zoom-Nikkor IF ED 18-135 mm f/3.5-f/5.6 G Lens
Nikon Autofocus Speedlight SB-800

Looking at the photo one thing is for certain you need more light.
Primary reason for dull image is the lack of light.
You are probably using a flash but since the photo is out in the open in the night the effect of flash may be reduced since the light does not have much to reflect off expect the subjects.
If you are not able to recapture this image try brightening the photo is some software, something this is the only way.

Try the following: (this is not in any particular order).
1. Increase the exposure setting. Over exposed image is better then a dark image. You can always make the image darker in some photo editing tool but an under exposed image cannot be effectively made brighter all the time.

2. If possible increase the brightness of the flash and increase the time of flash, what I mean is some flashes have a setting to make it brighter and/or increase the time the flash stays on. The aim is to make the light from the flash last a little longer so the camera gets enough brightness to capture the image clear.

3. Increase the ISO setting to a higher value. This will make the image grainy but it will be better then current image.
Change ISO as the last option. Best is to put it on auto and tinker with other settings first.

You will need to experiments few settings before you find the right one. It will be a combination of settings to make this work.

4. Change the position of the subjects.
a. Bring them closure to the camera for the effect of flash is more.
b. Place the subjects against a brighter background instead of open blackness. The empty black space is eating up the lighting.

5. Many camera has an Infrared light (IF). This light is thrown on the subjects to help the camera determine distance and exposure. Make sure this light is on and it reaches your subjects. If the IF light is not reaching the subjects then make sure it does, either you get close or bring them close to the camera.
This is critical, IF is designed for dull light shooting.

6. If your camera has a night shooting mode, use it. This is the best option.

What kind of camera are you using ?. Point and shoot or dslr.
Let me know how it goes.

I have a DSLR camera - Nikon D80 with an AF-S DX Zoom Lens 18-135mm, f/3.5-f/5.6
and yes, it has an IR light that comes on when it's dark, but I really thought that was used for autofocusing and not exposure settings, nevertheless, I now know, thanks to you.

Yes, my camera has nite mode, but I never used it. I always shoot in either P, S, A or M mode. It also has flash compensation where you can alter flash output by changing the level of brightness, but as you say, I need to fiddle around with these settings until I get it right.

Thanks for your comments, if I have any problems and need your assistance, I am going to look you up.

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Shooter
Nikon D80 DSLR Camera
AF-S DX Zoom-Nikkor IF ED 18-135 mm f/3.5-f/5.6 G Lens
Nikon Autofocus Speedlight SB-800