I’ve recently started a new position at a company and was introduced to everyone.  When I shook the hand of the head manager it was soft.  I was expecting a firm handshake and gave one myself.  I felt bad afterwards because it didn’t feel like we connected and I was worried that I hurt his hand.  Why would someone in a position of power have a soft handshake and what does this all mean?

That is a good question. Not knowing anything about the manager, my answer can’t go too deep. Yes, you are correct in guessing that the manager’s weak handshake suggests a lack of connection. The lack of connection is indicative of a lack of commitment. It sounds like your connection to the company is somewhat tenuous, and that you can be fired rather easily, at this point.

The manager’s weak handshake suggests a cow offering you its udder. It allows you to take a squeeze. There will, though, be a payment, for I suspect that the company will end up squeezing you, and milking your for all you are worth, both in terms of your knowledge and your time. My advice is too work very hard to prove yourself, but to still send out your resume to other companies. 

P.S. I am not, at all, suggesting that the same psychological motives are relevant for everyone who has a weak handshake. I am just saying that in this case it may be so, based on the particular workplace context, as you describe it.