SiS tourney A050 thumb 1x7 to either a 1x10 or 2x7
It's unclear whether you have drop bars or a flat bar. Normally 'roadbike' would strongly imply drop style handlebars, but you talk about 'horizontal' bars and are thinking about upgrading to Deore which is a mountain bike groupset. 'Integrated shifters' would normally mean shifters for a drop bar as mountain bikes have moved to separate shifter and brake units. However, the answer is the same.

If you have a 7 speed cluster of sprockets on the rear wheel, you either have an old-style 7 speed freewheel or a 7 speed freehub and cassette. See here for a description of the difference. In either case you will not be able to upgrade to 10 sprockets on the hub. Freewheels only go up to 7 speeds; the freehub body for 7 speeds is slightly shorter that those used for 8, 9 and 10 speeds and cannot accomodate a 10 speed cassette. To get 10 speeds in the rear you would need a new rear wheel with a 10 speed compatible freehub. You would then be able to fit a 10 speed road or mountain groupset. However, at this point you are looking at replacing more than half the bike, so it probably makes more economic sense to look for a used bike with a 1x10 or 2x10 drivetrain.

If you are prepared to stay at 7 speeds in the rear things are a bit easier. the Shimano Tourney line has 3x7 speed shifters for flat bars, Microshift makes Shimano compatible  3x7 road bike integrated shifters/brakes I believe. You could use derailleurs from Shimano's lines above Tourney as 7, 8 and 9 speed derailleurs all use the same actuation ratio. 

Obviously for 10 speed you'd need a 10 speed compatible chain, but if you stick with 7 speed I'd replace the freewheel or cassette and chain as well so you have a complete new drivetrain.