So far the information you have given as answers appears correct, there are a couple of minor points to note (in bold).
a. Design current is the expected load in watts divided by the supply voltage. You have stated 1900W/230V =8.26A
b. Based on the design current the circuit breaker must be rated higher than the design current, for a load of 8.26A the next highest standard breaker rating is 10A so the
minimum is 10A but higher can be used, also the characteristics of the load may require a type B, C or D circuit breaker. If the load does not show high current peaks on starting (or stopping) then a B type breaker is a good choice.
As the circuit is part of a swimming pool and assuming it is not SELV or electrically separated then it would require 30mA RCD protection as well.
c. The cable appears to be twin and earth cable installed in conduit on the surface, not a usual choice, but this would be installation method B, however BS7671 does not give tabulated current ratings for twin and earth in conduit, but insulated and sheathed multicore cable tables are pretty close.
d. from your answer the only rating factor that applies is an expected ambient temperature of 40°C (Ca at 0.91)
e. Determining the minimum csa for current carrying capacity means applying the rating factors to the In (16A in your case) as a divisor and referencing the appropriate table and installation method. With a corrected design current of 17.58A the minimum csa live conductor that can carry this load is 2.5mm² (for multicore cable thermoplastic insulated and sheathed).
Determining the minimum csa for voltage drop would mean a rearrangement of the usual equation to say:
mV/A/m (min) = (Max volt drop permitted * 1000) / (Ib*L)
Max volt drop for lighting is 3% of supply voltage, for a 230V supply this is 6.9V, Ib is 8.26A, L=46m
mV/A/m(min) = 6900/8.26*46 = 6900/389.16 =18.16 mv/A/m
From the tables of volt drop the next size of cable that has a volt drop <=18.16 is 2.5mm² as 2.5mm² has a volt drop of 18 mV/A/m.
f. The actual volt drop can now be calculated as you have done and got 6.83 V just within the 6.9V limit for lighting.
g. For 230V final circuits <=32A: on TN system earthing 0.4s, on TT earthing 0.2s; for distribution circuits and >32A circuits: TN system earthing 5s, TT earthing 1s.
h. earth fault loop impedance is calculated from Zs = Ze = (R1+R2) as you have done.
Because you have used resistance values at 20°C this would be equivalent to a measured value and so the maximum Zs you would refer to would be from the OSG and for a B16 this would be
2.2Ω (0.8* the BS7671 value to correct for temperature of the conductors).
i. the maximum earth fault loop impedance stated in BS7671 is 2.73Ω,
this is different from the value you have given (2.87) because it is taking into account the Cmin value of 0.95.
Overall pretty good, I think.