Getting lat/long for centroids using QGIS

Possible solution by means of PyQGIS. An Extension to the answer provided in this thread Adding coordinates to attribute table.

Use the following script instead

# imports
from PyQt5.QtCore import QVariant

# accessing point layer by name
layer = QgsProject.instance().mapLayersByName('test')[0]

if not layer.isValid():
    print("Layer failed to load!")

# specifying input/output crs and setting up constructor
crsSrc = QgsCoordinateReferenceSystem(layer.crs()) # source crs
crsDest = QgsCoordinateReferenceSystem(4326) # destination crs
transform = QgsCoordinateTransform(crsSrc, crsDest, QgsProject.instance()) 

layer_provider = layer.dataProvider()

# adding new fields
for attr in ["lat", "lon"]:
  layer_provider.addAttributes([QgsField(attr, QVariant.Double)])
layer.updateFields()

# starting layer editing
layer.startEditing()

for feature in layer.getFeatures():
    fields = layer.fields() # accessing layer's fields
   
    geom = feature.geometry() # accessing feature's geometry
    geom.transform(transform) # transforming feature's geometry
    feature.setGeometry(geom) # setting feature's geometry in a new crs
    
    attrs = {
            fields.indexFromName("lat"): round(feature.geometry().asPoint()[1],6),
            fields.indexFromName("lon"): round(feature.geometry().asPoint()[0],6)
            }
    layer_provider.changeAttributeValues({feature.id(): attrs})

layer.commitChanges()

Get the output that will look like

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References:

• QGIS Python API | Class: QgsCoordinateTransform
• How to make crs transformation of a QgsVectorLayer?

Inserting new fields with the field calculator and the expression $x and $y on the layer with your centroids should do the job. If you use the polygon layer of your lakes, you might include there lat/lon in the attribute table directly with x(centroid($geometry)) resp. y(centroid($geometry)).

If your layer is not in EPSG 4326, you can use this expression to reproject the coordinates and get lat/lon-coordinates (replace XXXX with the EPSG-code of your CRS):

x(transform( make_point (x(centroid($geometry)),y(centroid($geometry))), 'EPSG:XXXX', 'EPSG:4326'))

and

y(transform( make_point (x(centroid($geometry)),y(centroid($geometry))), 'EPSG:XXXX', 'EPSG:4326'))

Or even better: include the EPSG-code of your project with the corresponding variable, so you don't even have to know about what EPSG you are using and the result is always correct, even when changing the project-CRS. This expression should work without any changes with whatever vector layer you use:

x(transform( make_point (x(centroid($geometry)),y(centroid($geometry))),  @map_crs , 'EPSG:4326'))

and

y(transform( make_point (x(centroid($geometry)),y(centroid($geometry))),  @map_crs , 'EPSG:4326'))