You can travel abroad after applying for U.S. citizenship without any restrictions. However, you can travel only as a permanent resident by using your Green Card as you will become a U.S. citizen only after taking your oath of allegiance.
So you need to follow the rules applicable to permanent residents which require applying for a re-entry permit if the trip is for one year or more.
Can you travel abroad after the naturalization interview?
Even if you are done with your naturalization interview, you will only be a permanent resident until you attend the oath ceremony and receive your naturalization certificate. Hence, you can travel abroad as a Green Card holder, while your U.S. citizenship application is pending.
How travel can affect the naturalization application process?
Even though you can travel abroad there are ways that your travel can affect your citizenship application. All the laws that are applicable for permanent residents in order to travel will still be applicable.
Understanding the laws regarding travel and planning accordingly will help you to travel and at the same time not let it affect your naturalization application process.
While Form N-400, Naturalization Application is pending, you must fulfill the continuous residence and the physical presence requirement for the citizenship application. So your trip must not be lengthy.
At the same time, if you have not registered for updates by emails or text messages, you will have to ask your friend or your relative in the United States, to check your mails and inform you about your biometrics appointment and about your citizenship test and interview. You will have to see that you do not miss these appointments and missing your interview or your biometrics appointment will delay your citizenship application process.
Generally, it takes around 12 months or more to process a U.S. citizenship application but you are likely to receive your biometrics appointment notice within one month from the date of your application.
Hence it is better to travel abroad after you submit your biometric information. Some service centers accept walk-ins and if your USCIS service center accepts walk-ins, you can visit the application support center at any time and submit fingerprints and then travel abroad.
As mentioned before, until you take the oath of allegiance, you can travel abroad as a Green Card holder and return to the United States with your valid Green Card. Remember that, you will have to surrender your Green Card only at the time of taking the oath so do not lose your Green Card until then. If you lose it you should replace the Green Card.
It is recommended to remain in the United States during the naturalization process. But if you are required to travel abroad due to a genuine reason, you may do so, but remember to get back to the United States for your biometrics appointment and your naturalization interview.