Noob: 19v power supply without IEC (kettle) lead
For a mains voltage connection to a piece of equipment you would be well advised to stick to a standard connector type rated for mains voltage - especially if the device will ever be used by someone else or pass out of your hands. You cannot use a barrel connector for mains!

If the circuit or the case of your project needs to be earthed (grounded), you need a 3-pin connector, and the only standard one smaller than IEC C13 is the C5 'cloverleaf' style - but it's not much smaller. If you don't need an earthed supply you can use the C7 two-pin connector. Either way, you can look for an alternative enclosed PSU with the connector you prefer, or you can use an open-frame type and wire it to your own connector - as long as you understand and follow proper practices for mains voltages. Given the choice I prefer to mount an enclosed PSU so that the mains inlet is accessible through a cutout in the equipment case.

If you don't know how to decide whether your project needs earthing or not I suggest asking a separate question.