Yes. Refugees and humanitarian entrants may use public assistance, including cash, food, and medical, and still be eligible to get a green card or their citizenship. There is no “public charge” test for humanitarian entrants when they apply for their green card.
However, you should always give the government accurate and complete information when you apply for or get benefits. Immigrants who fraudulently get benefits may have trouble becoming citizens and may even face deportation if convicted of a crime.
You should also avoid traveling outside the U.S. for more than 6 months if possible, especially if you are getting cash assistance. You may have trouble re-entering the country. Always tell DSHS before you leave the country if you may be gone for over 30 days.
To learn more, talk to an immigration lawyer, or read Public Charge: What You Need to Know.