I'm implementing an algorithm into my Python web application, and it includes doing some (possibly) large clustering and matrix calculations. I've seen that Python can use C/C++ libraries, and thought that it might be a good idea to utilize this to speed things up. 
First: Are there any reasons not to, or anything I should keep in mind while doing this?
Second: I have some reluctance against connecting C to MySQL (where I would get the data the calculations). Is this in any way justified?
cython support for c++ is much better than what it was. You can use most of the standard library in cython seamlessly. There are up to 500x speedups in the extreme best case. 
My experience is that it is best to keep the cython code extremely thin, and forward all arguments to c++. It is much easier to debug c++ directly, and the syntax is better understood. Having to maintain a code base unnecessarily in three different languages is a pain. 
Using c++/cython means that you have to spend a little time thinking about ownership issues. I.e. it is often safest not to allocate anything in c++ but prepare the memory in python / cython. (Use array.array or numpy.array). Alternatively, make a c++ object wrapped in cython which has a deallocation function. All this means that your application will be more fragile than if it is written only in python or c++: You are abandoning both RAII / gc.
On the other hand, your python code should translate line for line into modern c++. So this reminds you not to use old fashioned new or delete etc in your new c++ code but make things fast and clean by keeping the abstractions at a high level.
Remember too to re-examine the assumptions behind your original algorithmic choices. What is sensible for python might be foolish for c++.
Finally, python makes everything significantly simpler and cleaner and faster to debug than c++. But in many ways, c++ encourages more powerful abstractions and better separation of concerns. 
When you programme with python and cython and c++, it slowly comes to feel like taking the worse bits of both approaches. It might be worth biting the bullet and rewriting completely in c++. You can keep the python test harness and use the original design as a prototype / testbed.
Using C/C++ for heavy calculations in Python (Also MySQL)