Possibly, yes. If the injection system is not providing a steady and regular supply of fuel to the engine, shuddering and hesitation could result. Note that the fuel pump is part of the fuel system, and can cause similar symptoms if it's failing.
Dirty fuel injectors or spark plugs
There are numerous issues that can cause a lack of power during acceleration. Common causes include a clogged fuel filter, a clogged air filter, and dirty fuel injectors.
Fouled, damaged injectors or incorrect timing on fuel injector pump
Sounds like clogged fuel injectors or fuel filter.
Cleaning the fuel injectors is a good start.
The primary reason for hesitation on acceleration from stop on a 2000 Toyota Echo is a dirty fuel system. This can include clogged fuel injectors or a dirty fuel filter.
Try a can of Seafoam Engine Tune. It will sometimes work wonders on clogged injectors. www.seafoamsales.com
Leaking injectors.
Normally you do not get carbon buildup on injectors. They can get gummed a little, or blocked, but that should not cause backfire. Check timing.
A malfunctioning fuel injector pump will cause the injectors to not fire properly. The fuel injector pump will need to be replaced.
Bad fuel filter or clogged fuel injectors
Normally it points to fuel not being entirely burnt in the cylinder, or too much fuel being delivered. Cause could be malfunctioning injectors, or some restriction in the air inlet, like a clogged air filter.