Assuming you know the distance and direction from a known point to each new sprinkler, QGIS has several tools you can use to map the new sprinklers.
Attributes toolbar > click on the arrow next to the button with a ruler > Choose Measure Line or Measure Angle
Allows you to measure lines and angles. Easy to use, but can only measure one at a time (lines OR angles).
Use the Measure Angle tool to find the correct bearing from the known point. (If you turn on snapping (in the snapping toolbar) it will be easy to start your measurements exactly from the known points.) Draw the new sprinkler point along that bearing. Use the Measure Distance tool to measure the correct distance. Move the new sprinkler point to the correct distance. Measure the angle again and adjust the point location. Repeat as necessary.
Advanced Digitizing Tools
Advanced Digitizing toolbar > enable advanced digitizing tools
Create a temporary scratch layer (lines), enable advanced digitizing tools, and start drawing a new line feature. The Advanced Digitizing window will display the angle and length of each line segment as you draw it. (d is distance, a is angle; x and y are the current coordinates under the mouse pointer)
Draw a line from the known sprinkler point to the location of the new sprinkler, based on the known distance and angle. Draw a point at the end of the line. If you turn on snapping (in the snapping toolbar) it will be easy to place the new point precisely at the end of the line.
Shape Tools plugin
Plugins menu > Manage and Install Plugins > find and install the Shape Tools plugin
Using the Azimuth Distance Digitizer tool, click on the known point. Enter the azimuth (compass bearing) and distance from the known point to the new sprinkler.
The tool will create a new point at the specified distance and direction from the known point.