Will a letter sent out saying we owe back taxes show up during the underwriting process in FHA home loan..we got a CP2000 letter?

So we did our 2014 taxes got a refund and did 2015 got a refund this year we started the process of getting an FHA home loan and everything has checked out so far but unfortunately right in the middle of underwriting we got a letter in the mail that's a CP2000 letter from IRS saying we owe 4K for taxes in 2014.... show more So we did our 2014 taxes got a refund and did 2015 got a refund this year we started the process of getting an FHA home loan and everything has checked out so far but unfortunately right in the middle of underwriting we got a letter in the mail that's a CP2000 letter from IRS saying we owe 4K for taxes in 2014. I double checked it's true because I did a typo error and on the wages part I put a 3 where it should have been a 5 and it was an honestly mistake..and we still ended up with a refund that year and got one this year...so it never even crossed my mind so I was shocked to get this in the mail! So here is my issue we have already come this far into the home buying process in fact we are about a week til a possible closing..( early according to the paperwork) we haven't responded to the IRS about this letter nor said anything about it we still have until March 16th to respond. My question is when they pull the transcripts will they see the corrected amount the IRS fixed according to their notice they sent us? Will it show we received this letter? We do not want to ruin our chances of getting a home over a stupid typo error. We are going to set up a payment plan and pay it off in full as soon as we close on the house but we don't want to say anything if it won't show up...you know don't ask don't tell. If we have to pay for this first then the money and work we put into this whole process will be lost and wasted. Thanks in advance.
Update: FYI UPDATE: WE PAID THEM BUT WHERE NEVER ASKED FOR TRANSCRIPTS AND WE TO MY KNOWLEDGE I DO NOT REMEMBER US SIGNING PERMISSION TO PULL THEM AT ALL BUT WE STILL GO CLEARED TO CLOSE AND EVERYTHING IS FINE...WE SIGN PAPERS MONDAY AND GET KEYS...SOOO BOOM! SHOWS HOW MUCH YOU GUYS KNOW...
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