Does CY8CKIT-001 Development kit support CY8C24423A-24PVXA Device?
CY8CKIT-001supports CY8C28 and CY8C29 families of PSoC1 besides PSoC3 and PSoC5. It does not support CY8C24x23.
For CY8C24x23, you will have to use CY3210 CY8C24x23 Eval Pod along with CY3210 PSoC Eval1
In general, PSoC1Kit Selector Guide seems pretty handy for finding right kind of kits.
You can refer to datasheet to get your queries