Shimano Claris 1x8 11-28t, can upgrade to 1x10 11-34t?

No. You cannot use a 10 speed cassette with an 8 speed derailleur and shifter.

Cassettes with increasing numbers of sprockets have narrower sprocket spacing, to allow the larger numbers of sprockets to fit on the hub. Shifters and derailleurs for different drivetrains with different numbers of sprockets are designed to move the derailleur cage the proper distance for each shift. Higher speed chains are also incrementally narrower to work with the narrower sprocket spacing. The chainrings are also more narrowly spaced to work with the narrower chain.

If you want to upgrade to 10 speed, you need a new cassette, derailleur, shifters and chain (and crankset if you were running a 2x setup). This is very often not cost effective vs simply buying a new bike with an upgraded drivetrain.