Hi Mike,
You could use StringReplace to delete all the characters including and after the space of the result like this:
hwd = Import["~/Desktop/hw.tsv", {"Data", All, 1}];
hwtemp = Import["~/Desktop/hw.tsv", {"Data", All, 2}];
date = DateObject[Rest[hwd]];
temperature = Rest[hwtemp];
temps = StringReplace[temperature, " " ~~ __ -> ""]
If you want to turn these temps into Integers, simply map ToExpression to interpret the number string as an Integer:
ToExpression /@ temps
If you need Real values rather than Integers, this would do the trick:
temps = ToExpression /@ StringReplace[temperature, " " ~~ __ -> "."]