One of the unintended side effects of the government shutdown is that the immigration courts are not funded. This delays immigration hearings which will be rescheduled after the government reopens.

As the article below explains, the average backlog for a hearing has grown from 400 days (just over a year) to 718 days (almost two years) in the past decade.

Despite Huge Backlogs, The Government Shutdown Halts Most Immigration Court Hearings

Hundreds of thousands await hearings in the already-congested system. Those delays could help some cases but hurt many others.

NPR.org

These delays and now rescheduling might have an effect in both direction. An asylum seeker, perhaps with a not very strong case, might stay even longer in the US before a judge denies an asylum request. An asylum seeker with a strong case might have have that case become less strong, e.g. if they were providing for a sick dependent who then dies.

trac.syr.edu