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My parsing of the following:

1. Antique public saw (8) OUTDATED (?)
public = out (as in outside)
saw the past tense of see, as in seeing someone meaning dating someone. Hence saw = dated

2. Fruit being supported by former statesman (7) PEARSON (?)
fruit = pear
cannot work out why supported = son
statesman = US American (someone called Pearson?)

Are the parsing correct?

The answers are what I think they are based on the grid and other answers. If they cannot be parsed, it may mean that my answers are wrong.

Thanks for any help.

1. Yes, your parsing is correct

2. If it's a North American puzzle, then the answer is indeed PEARSON (Lester B. Pearson, former Canadian Prime Minister), the parsing being PEARS ('fruit') + ON ('being supported by').

Thank you for your response. It helped. Thanks again.