IRS advice

So my husband hasn’t filed taxes in about 4 years. We’re looking to buy a house next year, and read that we will need the two most recent tax returns as part of the documents for a home loan. So far he filed 2 years and is on a payment plan. Now we receive a letter in the mail saying his w4 form is wrong. To my understanding, he was under withholding. With the un-filed tax returns and incorrect tax w4 form, are our chances delayed with becoming home owners ? What will happen if he doesn’t update the w4 form or file his taxes ?