The usual mooring at Whitby is on the pontoon above the swing bridge.
The depth varies as the river is dredged from time to time. With 1.6M you should be ok - mainly because you are likely to be 3 or even 4 boats out from the pontoon. Even if you touch the bottom is soft mud and you'll only be in it around low water - and you have to be much nearer to HW for the bridge to open to let you out.
Its well worth the visit - Whitby is a lovely old town, famous for its fish & chips, Abbey and Dracula legend.
Keep offshore towards the N Cardinal buoy before turning on the final line into the harbour then make sure to keep well to the West on your way in, the flood tide sets strongly west to east across the entrance , and if the wind is northerly best give the harbour master a shout (Ch 12) - it can get very nasty as you close the piers.
If you can bear to hear the truth you've spoken
Twisted by knaves to make a trap for fools,