idiomatic use of cut will be
cut -f2- input > output
if you delimiter is tab (“\t”).
Or, simply with awk magic (will work for both space and tab delimiter)
awk '{$1=""}1' input | awk '{$1=$1}1' > output
first awk will delete field 1, but leaves a delimiter, second awk removes the delimiter. Default output delimiter will be space, if you want to change to tab, add -vOFS="\t" to the second awk.
UPDATED
Based on your updated input the problem is the initial spaces that cut treats as multiple columns. One way to address is to remove them first before feeding to cut
sed 's/^ *//' input | cut -d" " -f2- > output
or use the awk alternative above which will work in this case as well.