This is a punishment?
One of my earliest memories is being young enough to be still sitting in a high chair on a Saturday with my Dad home. He worked two jobs and we usually only got to see him at dinner time. He liked hot, spicy food and usually had a jar of pickled jalapenos or hot giardiniera on the table. So, he gave me a whole jalapeno. I thought anything my Dad gave me was great. I was crunching away with tears running down my face and a big grin while my Mom was yelling at my Dad for giving it to me.
Cold showers were the norm as my Mom often hosed us down after playing outside before we were let in the basement to scrub off the second layer of dirt in the laundry sinks with Lava soap.
And thus we see the incredible power of context when it comes to reward vs. punishment.
As other posters have mentioned, assuming the hot sauce isn't spicy enough to cause severe pain and/or tissue damage, I don't know that in and of itself I'd consider it abusive. However, the context of its administration is VERY important--it's abuse if, as has been said, the mother administers the punishment vindictively and/or sporadically; conversely, if there are well-known and solidly-established rules, boundaries, and consequences, then it could potentially help in disciplining the child.
Given that the mother said she tried spanking first and it didn't work, though, I'd imagine the issue speaks more to ineffective parenting than unrelentingly-unruly children in need of steeper punishment.