I think you'd want to apply the alpha value of the sprite. You can do that by making the last line "COLOR = vec4(mix(...).rgb, texture(TEXTURE, UV).a)"
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This worked perfectly - thank you!!
For anyone else looking here later, the final shader code (confirmed working Godot 4.2) is:
shader_type canvas_item;
uniform sampler2D screen_texture : hint_screen_texture;
uniform vec4 water_color : source_color;
uniform sampler2D wave_noise : repeat_enable;
void fragment() {
vec2 water_wave = (texture(wave_noise, UV * TIME * 0.02).rg - 0.5) * 0.02;
vec2 uv = vec2(SCREEN_UV.x , SCREEN_UV.y - UV.y) + water_wave;
vec4 color = texture(screen_texture, uv);
float mix_value = 1.0 - UV.y;
float avg_color = (color.r + color.g + color.b) / 3.0;
avg_color = pow(avg_color, 1.4);
mix_value += avg_color;
mix_value = clamp(mix_value, 0.0, 0.7);
COLOR = vec4(mix(water_color, color, mix_value).rgb, texture(TEXTURE, UV).a);
}
Credits to Single-mindedRyan for creating this shader in the first place.
Oof, I haven't touched shaders in Godot myself yet, besides a very simple section during one tutorial, so maybe I should not give any advice, but I think this can be solved by applying the shader to the pond and adding a check for transparency before applying the changes to the pixels.