Hi I have a Dodge Neon 2003 with engine codes P0456 and P0440. What do these mean and how do I fix them

p0456 is a code for the fuel evapoation system so it sounds like the fuel tank breather system is faulty try running with the fuel filler cap off to help diagnose the issue

Google the codes to learn about them
& the system with a possible failure

How expensive a repair is, as you know,
will be told to you at a repair shop ,AFTER
you pay for the diagnostic process, to
find any issues

Is not a gas cap,could be clog cannister,low voltage [562],700 transmission control system.

p0456
Evaporative emission (EVAP) control system leak detected (very small leak)
p0440
Evaporative emission control system malfunction

Here you go
Read all about it
Looks like a fuel vapor leak and a missfire
http://www.obd-codes.com/p0456
http://www.obd-codes.com/p0441
http://www.obd-codes.com/p0303

p0440 - Evap. system performance (this could be something as simple as not turning the gas cap several times after you fill up.

p0456 - Evap leak monitor Small leak detected. - Could be related to the previous code.

p2106 - Throttle Actuator Control System - Forced limited power : This one you might have noticed the vehicle running rough, overdrive not working properly and cruise control will not set.

p2175 - Throttle Actuator Control System - Low airflow detected.

What you might want to try first is have the codes cleared which most auto parts stores offer this for nothing (unless you have a programmer then you can do it yourself). Pour in some fuel injector cleaner into your gas, put a new air filter on. And when you fill up your gas tank, crank the cap several times when you put it back on.
See if the codes come back on after that.

Buy a used engine from a reputable wrecking yard. That is the least expensive soulution. Then..........teach your girlfriend, where the dip-stick is!

replace your evap solenoid this should fix it.

a small evap leak is often interpreted as a loose gas cap , which it can be , it means the evap systems integrity has been breached , it can be an electrical solenoid , switch or just a vacuum line in the system . a bidirectional scanner can run the evap test after checking the gas cap , most well equipped shops can hook up a smoke machine to the evap system service port and run the smoke test , it there is a leak you will see the smoke , makes the diagnosis quickly . get an estimate from a reputable shop , by the way this problem has no effect on drivability , but should be fixed . cheers

P0456 = EVAP Emissions Control System Leak Very Small

Is the gas cap on tight or is the seal damaged? If it needs replacement, get it from the dealer.