Crossword Genius

Military staff protects old president, cycling this? (4-4)

Ross

I believe the answer is:

push-bike

I'm a little stuck... Click here to teach me more about this clue!

'military staff' is the definition.
The answer and definition can be both man-made objects as well as being singular nouns.
Maybe you can see a link between them that I don't see?

'protects old president cycling this?' is the wordplay.
I cannot really see how this works, but
'president' could be 'ike' (nickname of Dwight Eisenhower) and 'ike' is present in the answer.
This explanation may well be incorrect...

Can you help me to learn more?

(Other definitions for push-bike that I've seen before include "Bicycle (colloq.)" , "Vehicle driven by pedal power" , "Non-motorised vehicle" .)

I've seen this clue in The Independent.
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