If something is "at risk" it is exposed to danger.
"At stake" has a slightly different inference. The expression is adapted from gambling terminology. Literally, a "stake" in gambling is the money gambled on an outcome.
Metaphorically, when something is spoken of as being "at stake" the inference is that, like a gamble, the particular thing has been deliberately placed at risk in the hope of "winning" rather than losing; whereas many things can be "at risk" for other reasons.
That building is at risk of falling down.
Nobody deliberately builds something knowing it has a chance of falling down. The risk here has likely come about through decay or some other causes over time. The risk to the building has not been deliberately brought about.
He placed his job, his reputation and possibly his marriage at stake.
If someone deliberately places something at risk hoping that the risk will not materialise they are often said to have placed it "at stake", as if they have taken a gamble.
In your example, it sounds like either could be used, but I feel that they would need to be reworded slightly:
He has put his administration at stake on these promises for reform coming through.
He has staked his administration on these promises for reform coming through.
When you deliberately take a risk, the "stake" is there from the start, not only if you lose, so the conditional clause in your statement shouldn't really be there.
His administration is at risk if he doesn't come through on these promises for reform.
Even if it was deliberate and placed at stake, the administration itself is at risk and so it is possible to use this word in this way with the conditional clause..