Hi there, I'm not a financial professional, but I have some observations/advice from my experience.
I've been told that an overdraft is not viewed the same as a loan or credit card, but can be regarded as the worst form of credit, and it was recommended that I do not use my overdraft as it reflects that I can not manage my money.
It is better for your credit rating to use your credit card and pay it, than to use your overdraft facility.
A question would be - Have you checked your credit rating on Experian or Equifax or are you assuming it is not good? Just having no debt does not equal perfect credit rating, and often not having credit cards or agreements that you can be seen to honour gives you a lower score.
Your loan should be fine for the mortgage applicaiton (as long as you have been paying it and it is affordable), all that will happen is that it will be deducted from the amount of money that you could potentially borrow.
I would personally pay off the overdraft with some of the deposit, leaving you with 16K.
This would leave you with needing 134K, which is still a 90% or less mortgage. It makes little difference to the deals available (on the 90%, 80%, 75% etc) that you don't have 20K.
What it can make a difference on is how much you can borrow.
If you need a 134K mortgage based on a multiple lending ratio of 2.5 : 1 (Multiples of your joint income), you would need to have a joint income of 54K a year.
Given the 4K loan remaining you would need this to be 55,200 a year for a lending of 138K (debt capacity) - 4K loan leaving a mortgage capacity of 134K
You should use the mortgage comparison tools to see what the mortgages are available for this (there are some available).
With some mortgages it is possible to port the mortgage to a new home, so it is worth contacting the Halifax to see if they would be able to do this. This can be a quicker and less painful approach. Don't forget to ask about the cost of additional lending (if the new mortgage is greater than the previous one) as it might be higher, and also about the fees (to port, or to apply for the new mortgage or additional lending).
Good luck with it all.
BTW paying your mortgage for 4 years should look good on your credit record so chin up! :)
Sparky