Sometimes asylum cases take a very long time to process. The initial interview may be a year or more away. Asylum applicants sometimes get ansy about remaining in the U.S. for so long and ask us whether it would be okay for them to leave the country.
If you have an asylum case pending and are thinking about leaving the U.S., you need to think about a few things.
USCIS does not explicitly prohibit asylum applicants from leaving the U.S. while their application is pending. However, you must file an I-131 application for advance parole and have that application approved BEFORE you leave the U.S. If you leave the U.S. without obtaining advanced parole, you will most likely be deemed to have abandoned the asylum application and may not be able to reenter the U.S.
You should file your advanced parole application at least three months before you intend to travel. It usually makes sense to request advanced parole shortly after applying for asylum as you never know when you may need to leave the country quickly.
Leaving the U.S. should be avoided if at all possible. Even with a pending asylum application, if you have inadmissibility grounds - such as a visa overstay or criminal history - you may not be allowed to return.
Finally, the most important thing about asylum and travel is that you CANNOT return to the country that you are claiming in the asylum application would persecute you if you were to return. This would not only be fatal to the asylum application, but it would also open you up to a possible claim of immigration fraud.