You can use both. 'A little' is more formal I suppose. You can say 'a little bit' too. I think it would be a better sentence if you use ' (-) shorter'.
AE she had her hair cut (a hair cut) as Florentia said either bit or a little. Had it cut means someone (a salon) cut it. She cut means she did it herself. Some might say she cut also.A very common expression she had a trim or she had her hair trimmed.