The LCD is an electronical independant part and does (as long as there is power supplied) what you want. When you tell it to display something, it will. And if you tell it to display space characters (" ",0x20) it will do as well. so write a routine that clears (with space-characters) the two display-lines or write "Hello Customer" to the display. Afaik there is no explicit clear-function and I did not test if another LCD_Start() will work.
As told by Bob, the Character LCD is a dumb thing that does what you tell it to do. Once you print something on a char LCD(which is nothing but writing in to the data registers of the LCD) it gets latched and if the electronics unit(in this case PSoC) that is dricing it turns off the data is still held as long as the LCD does not lose power. There is LCD_Clear() routine that clears the char LCD. Open the Char_LCD.h file to gain access to APIs such as LCD_Clear() and more.