I have single-end smallRNAseq data and I have to trim the adapter (adapter sequence: `AGATCGGAAGAGCACACGTCTGAACTCCAGTCAC`). After trimming the adapter using the bbduk command mentioned below, I checked the reads in my data and they still have the overhangs of the adapter as shown below.

The adapter overhangs can be seen below:

    cat bbduk_trimmed_small_RNA_001.fastq | grep TCGCAGGGAAATCATCTGATTA

    TCGCAGGGAAATCATCTGATTAGATCGGAAGA
    TCGCAGGGAAATCATCTGATTAGATCGGAA
    TCGCAGGGAAATCATCTGATTAAGATCGGAAGAA
    TTCGCAGGGAAATCATCTGATTAGAA

or can be seen here:
https://postimg.cc/image/knuvtyv0d/

This trimming was done using bbduk.sh (bbmap tools: https://jgi.doe.gov/data-and-tools/bbtools/bb-tools-user-guide/bbduk-guide/). Here NEB-SE.fa file has the adapter sequence mentioned above.

    bbduk.sh -Xmx1g in=small_RNA_001.fastq out=bbduk_trimmed_small_RNA_001.fastq ref=NEB-SE.fa ktrim=r k=13 mink=6 minlength=18 hdist=0

I could reduce the kmer size `k=4`, but that would risk into trimming the false positives. How can I completely get rid of these adapter sequences from my data without trimming false positives? 
You can't have it both ways. You can either be strict about trimming and risk losing a few reads or have a few reads left that may have a bit of adapter. Your aligner should deal with any extraneous bases while it does the alignment. Have you tried to align this data to see what fractions aligns? With smallRNA you are looking for a specific length (~21-25 bp). If you reads are failing that length criteria after trimming you may want to discard them since they may not be what you are looking for.

BTW: I am curious as to what the original reads in the example you included look like (or is that the full original sequence)?
