Learn English – word or term for letting someone else have a chance to win

This can be described as role-projection. I.e. "imagining oneself in the role of another".

This usage isn't common, but it exists in the psychology literature. E.g.:

Hirschman (1983) and others (d'Astous & Deshenes 2005) posit that certain traits can influence participation in fantasy-related consumption behaviours, including role-projection and escapism (i.e., to get away from one's everyday life) or other activities involving visualization. Specifically, an individual's preference for using imagination, can predict a tendency to engage in a certain types of fantasy, including both role-projection and escapism.

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Imaginal Responses mainly refer to role projection and escapism.

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"Extending your hand" is indeed correct, understood exactly in this way, and of long use for precisely this. At least one dictionary even lists "extend your hand" as a phrase with precisely that meaning. The OED doesn't, but does have a definition:

To stretch forth (the arm or hand). Also, to hold out, put forward (a staff, etc.).

And includes examples of Shakespeare and Dryden using it in this way.

All in all, you couldn't ask for a more apt word choice.