Can anxiety make it feel like your airways are constricted and make it difficult to breathe? like a clicking sound at the back of throat?

Yes... possibly.: No question that anxiety/panic attacks can have an associated symptom of shortness of breath. It's often a diagnosis of exclusion. I'd want to make sure you weren't having any underlying asthma, etc causing your shortness of breath (getting pulmonary function tests) and also asses your anxiety problems before hanging my hat on your dyspnea as a panic attack. Face-to-face eval should be done.

Anxiety and asthma: It is unsure which one causes which, anxiety or asthma. However, you should see a doctor and have a lung function test done to make sure you don't have an underlying untreated asthma. Anxiety by itself can causes multiple subjective symptoms including the ones you list above and generally they are benign since they do not have any organic causes.