I find myself riding with locked elbows most of the time on my Specialized Sirrus Elite. Probably too much pressure on the hands. How much of this is due to a weak core? Does a strong one really make a difference and a more comfortable ride? I've followed established bike fitting procedures. I'm getting about 10 miles of riding a week in the winter.

Lars

Without seeing a pic of the bike (preferably with you on it), it sounds like that you've adopted too aggressive of a posture.

It sounds like you are riding in an agressive riding position leaning forward and having to lock your arms to hold your upper torso up.
That is fine for being aero and speed but unless your racing that isn't the most comfortable way to ride.
Your elbows shouldn't be locked, but should have a slight bend at the elbows, and leaning forward will put pressure on your hands, shoulders, neck, and wrists, which can cause numbness in your hands over a period of time in that position.

You are saying that the bike is fitted but it sounds like bringing the bars up and back and possibly sliding the seat forward a bit might help and put you into a slightly more upright and comfortable position.
If sliding the seat forward doesn't help you can buy a shorter stem with a more upright angle or even an adjustable stem.

I had trouble finding just the right angle and comfort level on my hybrid and changed bar angle and added bar ends, but the best thing I did was add a post extension which brought the bars up about 3" and it cured the discomfort I was experiencing making my bike a pleasure to ride.

I'm no racer, and riding a bike in too aggressive a position, which I didn't need, dosen't lend itself to comfort no matter how many hand positions you have.