You seem to be confused about the terminology. A typical CPU core is an ASIC.
An ASIC (application specific integrated circuit) is exactly what it says on the tin - an integrated circuit designed to perform one function - in this case, executing machine code. Most of the chips you see inside electronics are ASICs (everything from operational amplifiers to ethernet PHYs and CPUs).
I don't usually see the two terms used together, because of the "application specific" vs "general purpose" thing. So I thought there might be something I was missing.
FPGA is more "general purpose" because you can reconfigure it for almost any purpose. A CPU, compared to it, is more "application-specific", though not as specific as e.g. an ASIC for handling crypro.
I probably added too much contrast into my original comment.
An ASIC (application specific integrated circuit) is exactly what it says on the tin - an integrated circuit designed to perform one function - in this case, executing machine code. Most of the chips you see inside electronics are ASICs (everything from operational amplifiers to ethernet PHYs and CPUs).