Is it possible to repair a Toshiba external drive?

The only way to recover data from it when the head has been knocked out of alignment is to find a data recovery place. What they'll do is open your drive and then copy the information that they can find to some form of secondary medium like DVDs, flash drives or an external drive. It's going to cost a specified amount per GB of data they find which is normally around $500-$900 depending on what they can recover.

So if it's worth that amount of money to you then go for it. Otherwise cut your losses and be more careful with your external drives.

When you dropped the drive the Read Arm hit the disk and damaged it.  There may be a chance to recover the data but it will have to be sent to a place that does data recovery and it is expensive.   As a Replacement get a Solid Sate Hard Drive (SSD).

External/portable SSD/s are available.

Yes if you have the moneyb

Yes but at very great Expense.

Clicking on a hard drive means it is way broken.

There is no repair for this. Only the hope you have back ups of what was stored on it.

Try a different computer first,

Take the drive out of the case and try it bare with an adaptor. 

I have had some luck with freezing the drive for 24 hours and trying again Can't hurt and I got several hours of use from a frozen drive before it failed again. Don't know why this sometimes works. 

Restoring the data on a clicking drive can be costly, much more costly than a good backup plan.

The only thing you can try is to drop it again to see if you can get it to function. You've nothing to lose since it's not functional at all. It sounds to me like the read/write head that scans back and forth over the spinning disc has crashed into the disk and is currently jammed the only way to possible un-jam it would be to give it another hit either with something like the wooden handle of a hammer, or drop it again. It's possible that another physical shock like it already sustained might just free up the head. The clicking is the read/write armature is actuating but it is not moving. You've got nothing to lose give it a few hits and/or try dropng it again.

They have data recovery labs you mail the drive into and they dismantle the drive and send you the data on a new drive.   Expect to pay a few hundred dollars.  Otherwise you can't repair it on your own. 

your local computer shop can tell you if it has been damaged beyond repair. Note that paying a data recovery service is quite expensive

unless you have $$$ it's fubar

That would depend on how valuable the information on the drive is and what you are willing to do or pay to be able to access the drive again.  You could start by looking the drive up on the Toshiba website and see what support options are listed.

If they are unable to help you could always contact a local computer repair shop and discuss the issue with them.