Firstly, don't be so hard on yourself. Being good at x, is different to being good at doing x as a freelancer. You (as we have all done/are still doing) need to learn all the other stuff that comes with being a freelancer. My estimates can still be out, two years along and I work with other freelancers who say similar. I want to do a good job, so sometimes it means I over deliver.
I personally would contact the person who commissioned you (rather than another at the business, that could back fire I think) and:
- apologise (if you didn't when you spoke)
- offer to put it right (again if you didn't already/if it isn't too late)
- ask for feedback once the updated document is with them
At best you want to walk away with a reference, at worst at least you know you will have tried your best to rectify the issue. You don't want this denting your confidence going forward.
Try to think of practical things you could differently next time, eg:
- Purchase the software you mention - if that possible
- Write notes from the meeting when you are commissioned and feedback that back in a proposal to help clarify what you are being asked to do
And maybe try to limit who you talk to about this within your world. Bad experiences and gossip can travel. Not sure if you'd mentioned what had happened to the other freelancer etc, can you trust them etc?
I would also say, that some clients aren't the easiest to work with and perhaps this inital company isn't for you in the long term. It sounds like it was you that made the mistake, but it could easily have been two sided. Was your contact really clear etc? Maybe not....
It is a strange idea that you don't want all that work that is out there, but early on I had a bad experience with somebody and the wise people on this board said 'dump and move on'. It was a revelation!
Re the meeting next week. If they want to know how the project for this first client went you could say it is still on-going. But focus on your job related experience and what you have prepared to talk about for the meeting. They are obviously vaguely interested if they are taking the time to meet with you, I am sure you recall from working days I rarely met somebody I had no interest in. Try to put this behind you, you have a chance here, with a different client to do a good job, get a reference and get going after the first "blip".
HTH