Can a cell start a mitosis if it lacks energy or molecules to complete it?
The short answer: no. First, let's get an understanding of the cell cycle control system, as there are some important molecules involved in this system that regulate mitosis. Think of the control system as a series of stoplights: as you mention, there is one stoplight at the G2 phase. There are two additional checkpoints: one at G1 and one in the M (mitotic) phase, just before Anaphase. Why before anaphase? At this stage in mitosis, the cell checks to determine if all of the kinetochores of the duplicated sister chromatids have microtubules attached to them. This is to ensure proper distribution of chromatids in anaphase.

Ok, so back to your question: what are some of the molecules involved, or what are some molecules that the cell might need before it proceeds?


Cyclins - these are proteins that are produced in the late S-G2 phases of Interphase. They activate cyclin dependent kinases (Cdks)
Cyclin Dependent Kinases - as the name suggests, these proteins (kinases) are dependent on cyclin in order to operate. The best known Cdk is called MPF, short for maturation promoting factor.


Throughout the cell cycle, these Cdks, together with their bound cyclins, are able to phosphorylate various proteins in the cell as it undergoes mitosis. For instance, the MPF would phosphorylate lamins, which are proteins on the nuclear envelope that can initiate degradation of the laminar matrix. This is essential in Prophase, when the nucleus is disassembling. There are many other things it can phosphorylate. For a more complete list, look under "Targets of MPF" in the above link for MPF. Take a look at this diagram, taken from Campbell's Biology (9th Ed):



There is one other molecule worth mentioning:


Growth factors - these are proteins specifically released by neighboring cells (relative to the cell that is preparing to undergo mitosis). These are crucial for mitosis to even initiate. A great example of a growth factor is PDGF, which stands for "Platelet Derived Growth Factor". These PDGFs are released by platelets at sites of injury, such as a cut on your knee. The PDGFs stimulate cell division and help your skin to reheal!


There are many other essential nutrients that cells require before they can initiate mitosis; not all have been listed here. Also, I know I didn't necessarily answer the "energy" part of your question, but I hope this helps for your first part.