As you describe it, you have a potential divider with an adjustable 10..20K pulling up to 5 V and a Light-Dependant Resistor (LDR) pulling down to 0 V. In parallel with the LDR is the base-emitter of an NPN transistor.
The base-emitter of the NPN will act like the anode-cathode of a diode. This stops your potential divider from dividing potentials properly. When the divider output goes beyond around 0.6..0.7 V, this diode will be forward-biased and will conduct current from its anode to cathode or, in your case, from base to emitter.
You would be better off using a comparator with hysteresis to detect your potential divider voltage and therefore your trip point. You can select one with an input current far less than your potential divider conducts at lowest light and it won't significantly disturb your potential divider's accuracy. The comparator output can drive your transistor's base with a suitable pull-up or series resistor, depending on the actual comparator you select.
(Descriptions of comparators with hysteresis are readily available on the interweb. If you're confused by any of it, post a drawn (or even photo-of-sketch) schematic of what you have and I can help you progress it from there. Schematic editor here is surprisingly dead easy.)