I'm just asking here whether doing batch correction in a design with 3 batches where one of the batches only contains one sample is possible. From this post Batch Correction with only one sample in a batch#:~:text=The%20bottom%20line%20is%20that,that%20sample%20is%20completely%20uninformative it seems like that sample would just be removed from the analysis, however there isn't any messages of that while running Deseq so I'm just asking for clarity whether the sample is removed or kept in the analysis.

This is the experiment design

BATCH   SEX RESPONSE
1   F   A
1   F   A
2   F   A
2   F   A
2   F   A
2   M   A
1   F   B
1   M   B
3   M   B
2   F   B
2   M   B
2   M   B
2   F   C
2   M   C
2   M   C
2   M   C
2   M   C

And this is the deseq design

dds <- DESeqDataSetFromMatrix(countData = count_matrix,
                              colData = coldata,
                              design = formula(paste("~", " batch", "+", " sex", "+", comparison_variable)))

dds <- DESeq(dds)
DE_output <- lfcShrink(dds, coef=paste0(Condition,"_",A_vs_B), type="apeglm")

That one sample won't be useful at all. As Ryan Thompson explained way back in the day, the coefficient for that sample will be the difference between that sample and the baseline sample type. Since there is only one such sample, that is equivalent to subtracting the sample out. In addition, your response 'C' is nested in batch 2, which means that batch and response are not orthogonal. You would probably do better by using the limma-voom pipeline to fit a mixed linear model instead.