Dont work directly under a lamp. Lighting conditions will change the look of the replacement pieces, and its difficult to achieve a match that will stand up under all lighting conditions. Also, if you are comfortable with Paintbrush or comparable art programs, you might consider using them to recreate artwork for replacement piece and skip steps below.
First put small pieces of cardboard in hole where piece missing to raise up replacement piece.
With paint brush and with blank piece outside puzzle on paper towel, wet blank piece with water followed by small amount of Gum Arabic to hold the color and add some gloss.
Study color areas and detail of surrounding pieces and decide what colors and detail will make replacement piece blend in best.
Color blank piece by extending color areas of surrounding pieces starting with light colors first. Use pencils wet as you mix colors; brush mixes colors well. Dont worry about detail at this point.
Dry with hair dryer about 10-12 away; remove small pieces of cardboard in hole and insert dried replacement piece.
Using pencils dry or slightly damp, carefully go over color areas with appropriate color pencil to bring it in line with surrounding pieces. Intense colors may be obtained using a tiny amount of gum Arabic and dark pencils, but the gum Arabic adds a gloss. Going over it with a dry pencil will lessen the gloss and the intensity. Work with it until you reach an acceptable balance between gloss and color intensity.
Now add detail with dry watercolor pencils. To add more color, dampen pencil on wet sponge; to intensify color use tiny amount of Gum Arabic on pencil point.
If not satisfactory, turn piece over on a paper towel, wet and blot and start again.
Let dry overnight; retouch next day to improve match as best you can, balancing dry and slightly damp pencils and tiny amounts of Gum Arabic for dark areas. The more you work an area, the more gloss will appear; to reduce, use dampened finger.
If the replacement is an edge piece, use dry or damp watercolor pencils to color the edge to match adjacent edges.
Use flesh pink and light brown watercolor pencils to line the top edge, simulating the paper wear around edges of the other pieces in the puzzle.
Check height and fit of replacement piece; often, use of water will slightly expand piece and require refitting.
IMPORTANT: Sign the back of the replacement piece with your initials, month, and year.