Great work on doing a little bit of troubleshooting to start out with! I've got a couple of articles that I believe will help you with your issue. If you are having issues with specific songs that will not play on your iPhone, then you will want to refer to this article (the article is written for iPods, but will definitely work the same for your iPhone 4):
Specifically, you may want to follow the instructions listed below:
Make sure song files are not damaged
If a specific song skips when you play it, it may be damaged.
1. Delete the song and update your iPod.
2. Insert the CD you used to import the song and import it again. If the CD is scratched or dirty, iTunes may not be able to import the song. Try cleaning the CD or use a different one. If the song was imported or created using a third-party application, try the following:
- Use iTunes to import the content directly from the CD.
- Convert the content to an iPod-compatible format using iTunes.
- Update your iPod with the new copy of the song.
Another step you may want to try can be found in this article, concerning issues with applications purchased from the App Store:
The step you are looking for specifically will be this:
If the issue affects only a single app, try closing just that app:
- Press the Home button to return to the Home screen.
- Double-tap the Home button to display recent apps.
- Tap and hold the affected app until the red minus appears.
- Tap the red minus to quit the app.
- Press the Home button, then restart the app.
If these steps do not help resolve your issue, you may want to visit getsupport.apple.com to get in touch with an Apple technician who will be able to help you resolve your issue. Thanks for using the Apple Support Communities!
Try a reset: Simultaneously hold down the Home and On buttons until the device shuts down. Ignore the off slider if it appears. Once shut down is complete, if it doesn't restart on it own, turn the device back on using the On button. In some cases it also helps to double click the Home button and close all apps from the tray before doing the reset.
Also - you might consider deleting and resyncing (or re-downloading) the music tracks.