What's actually happening
Your HVAC short cycles after an outage because the power interruption left something in a fault state. The system tries to hits the unresolved condition, and shuts down within a minute or two. Then it tries again.
The good news: most of the time nothing is broken. The outage just caught your equipment mid-cycle and it needs a clean, full power-down to reset.
The reset most people do (flipping the breaker for two seconds) isn't long enough. Control boards hold a charge and keep their fault memory. You need a full 5 minutes with power off.
The bad news, when it's bad: a spike on restore can cook the 24V transformer or a component on the control board. That's the expensive path, and you can usually smell it or spot a blown fuse.
Why it happens, most likely first
The single most common cause is the thermostat. Older programmable units and any thermostat with weak backup batteries can lose their brains during an outage. Some come back in an aggressive recovery mode and fire quick bursts that look like short cycling.
Right behind that is a control board safety lockout. When power dies mid-cycle, the board latches a fault it won't release until you force a full power-down. It'll keep trying and failing until you break the loop.
Then come the safety switches. An AC compressor that lost power under load can trip its high-pressure switch. A furnace that stopped mid-burn can trip the high-limit if heat pooled in the cabinet. Both cause fast cycling until pressures equalize or the cabinet cools.
On the heating side, a marginal flame sensor is a classic post-outage failure. The furnace lights, can't prove flame, and shuts down in 5 to 10 seconds. It retries a few times, then locks out.
The least common but priciest cause is surge damage to the transformer or board. If power came back with a spike, it can blow the low-voltage fuse or scorch a component. This is where the burnt smell comes from.