Does old/low charge battery affects internal clock?
Let's assume by "internal clock" you mean the internal oscillator, as the arduino's processor does not have a built-in real-time clock (something that is meant to keep track of real time like seconds/minutes/hours.)

The arduino is not like a motor where it runs slower if a lower voltage is applied. If the voltage goes too low, the arduino will just reset. Once it resets, the program running in it just starts over again, unless the battery voltage is so low it can't even start up.

If you're feeding it with a 9V battery, you're going through the 5V linear regulator on-board, which is not very energy efficient. Keep that in mind if your goal is to create a project that is meant to run a long time on the 9V battery. See the notes at http://www.gammon.com.au/power for tips on reducing the power used by an arduino and/or ATMega-based project.