Extract the vertices of the line and than delete all the vertices that are very close to one of the points. Than re-connect remaining vertices to a line again.
Your line layer needs a field with a unique field, in my case id. Layer names: lines (black line on the screenshot below) and points (red dots). Change all these names accordingly.
How you do it:
Menu Vector / Geometry tools / Extract vertices: white dotsOn the resulting layer from one, select features using the toolbar icon
select by expressionwith this expression. It looks for the nearest feature from the layerpointsand if the distance is smaller than 0.1, it is selected (in fact, distance should be 0 - but to be sure in case of rounding errors, set a minimal distance):
distance ( $geometry, array_first ( overlay_nearest( 'points', $geometry ) ) ) <0.1 Delete the selected points (yellow in the screenshot).
Menu Processing / Toolbox / Points to Pathto re-connect the remaining points: for sorting, select the fieldvertex_index, for grouping selectid. You get the blue line, corresponding to the original, black line, minus the vertexes where your have red dots.
The result looks like this. Be aware that the topology changes: on the T shape at the right, be deleting the vertex where both lines meet, being in the same time the start point of the vertical line, you loose the first segment and thus, the two lines are no longer connected:
