My TV has sound but no picture. Also, I periodically have to unplug and plug back in the power cord to turn the TV on. Model number: 42HF7945D/27 Serial number: YA1A0749019584 Can it be fixed?

Fault in the main power supply regulator board. Check and replace damaged component/s at its main power regulator [SMPS] section circuit. Contact any service technician. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts.
http://electronicshelponline.blogspot.com/
It will be best to replace the power supply regulator board as card basis rather than making "Surgery" to it; which usually will make loss for both money and time. Even though you can make it right, it will cost around an amount equal to the cost of a new board; including the cost of (spare parts + service charge) More to say, most of the spare parts to it will be difficult to get from open electronic spare part market. This is true to most of the flat panel TV power supply boards; irrespective of its brand and screen size.