I suspect you'll have no trouble at all.
My wife & I recently flew to the US from the UK with a brand new (and very cheap) suitcase with a front zip-up pocket. It got to the US okay, but on the next (domestic US) leg of the journey, the front panel had been torn off - it was hanging by a scrap of fabric and the contents of the case were protected only by the inner lining of the bag. We patched it up with a lot of very obvious bright-blue duct tape, and you could see at a glance it had been repaired. As I recall, the airline didn't even comment when we checked it back in for the flight home; it certainly got accepted okay.
The airline might want to make a note that it's already damaged (in case you tried to claim for damage) but it seems very unlikely they'd refuse to carry it unless it looked totally unsound (eg tied up with a single bit of string) or if it might be unsafe for baggage handling machinery (or handlers). For example, imagine a roller suitcase where the extending handle is jammed "up" - and so it permanently has a two-foot metal pole sticking out of it. I can imagine an airline might reject this.