Should Governors give Tax breaks to company's?

It depends. For example some states give commercial property tax breaks, and/or corporate income tax breaks which reduces revenue from those sources. But at the same time that encourages the businesses to operate in the state, hire employees who all pay income taxes, sales taxes, and property taxes (even renters rent includes property taxes) , offsetting the lost corporate tax and.commercial property tax revenues, with even more tax revenues raised from the employees it hires

When companies move out in a couple of years, it is generally because the tax beaks they were receiving in the previous state, were "sunset" taxes, meaning they were temporary. Successful states offer the tax incentives as permanent tax break to prevent that.

Governors do not have the power to "give" tax breaks. That is up to state legislators and the governor to sign the bill into law. Furthermore, yes they should give companies tax breaks to keep jobs in state. If they drive a manufacturer out of the state, then they'll lose a lot of money in sales tax and income tax.

various states do this. so, the companies go shopping to get the best deal.