Are these series truly missing? BBC iPlayer has the current series (2023) numbered 53, but the program's website only lists 38 of them (not counting specials).

Whereabouts are you seeing '53' on iPlayer?

Metadata retrieved with get_iplayer. Not sure exactly where it gets the information, apart from "channel schedule pages on the BBC web site".

Here is the output for all Imagine... episodes currently available in iPlayer:

imagine.s09e04.20060613.A_Picture_of_the_Painter_Howard_Hodgkin_.xml-   <series>Spring 2006</series>
imagine.s09e04.20060613.A_Picture_of_the_Painter_Howard_Hodgkin_.xml:   <seriesnum>9</seriesnum>
--
imagine.s20e01.20091117.The_Year_of_Anish_Kapoor_.xml-  <series>Winter 2009</series>
imagine.s20e01.20091117.The_Year_of_Anish_Kapoor_.xml:  <seriesnum>20</seriesnum>
--
imagine.s38e05.20141118.Anselm_Kiefer_Remembering_the_Future_.xml-  <series>Winter 2014</series>
imagine.s38e05.20141118.Anselm_Kiefer_Remembering_the_Future_.xml:  <seriesnum>38</seriesnum>
--
imagine.s41e01.20160719.DANGER_Cornelia_Parker_.xml-    <series>Summer 2016</series>
imagine.s41e01.20160719.DANGER_Cornelia_Parker_.xml:    <seriesnum>41</seriesnum>
--
imagine.s43e01.20161029.The_Seven_Killings_of_Marlon_James_.xml-    <series>Autumn 2016</series>
imagine.s43e01.20161029.The_Seven_Killings_of_Marlon_James_.xml:    <seriesnum>43</seriesnum>
--
imagine.s44e03.20170306.Alice_Neel_Dr_Jekyll_and_Mrs_Hyde_.xml- <series>Winter 2017</series>
imagine.s44e03.20170306.Alice_Neel_Dr_Jekyll_and_Mrs_Hyde_.xml: <seriesnum>44</seriesnum>
--
imagine.s49e01.20190714.Faith_Ringgold_Tell_It_Like_It_Is_.xml- <series>2019</series>
imagine.s49e01.20190714.Faith_Ringgold_Tell_It_Like_It_Is_.xml: <seriesnum>49</seriesnum>
--
imagine.s50e01.20200102.Lenny_Henry_Young_Gifted_and_Black_.xml-    <series>2020</series>
imagine.s50e01.20200102.Lenny_Henry_Young_Gifted_and_Black_.xml:    <seriesnum>50</seriesnum>
--
imagine.s50e02.20200723.Lemn_Sissay_The_Memory_of_Me_.xml-  <series>2020</series>
imagine.s50e02.20200723.Lemn_Sissay_The_Memory_of_Me_.xml:  <seriesnum>50</seriesnum>
--
imagine.s51e01.20210328.Kazuo_Ishiguro_Remembering_and_Forgetting_.xml- <series>2021</series>
imagine.s51e01.20210328.Kazuo_Ishiguro_Remembering_and_Forgetting_.xml: <seriesnum>51</seriesnum>
--
imagine.s51e01.20210902.Bernardine_Evaristo_Never_Give_Up_.xml- <series>2021</series>
imagine.s51e01.20210902.Bernardine_Evaristo_Never_Give_Up_.xml: <seriesnum>51</seriesnum>
--
imagine.s52e01.20220207.Marian_Keyes_My_not_so_Perfect_Life_.xml-   <series>2022</series>
imagine.s52e01.20220207.Marian_Keyes_My_not_so_Perfect_Life_.xml:   <seriesnum>52</seriesnum>
--
imagine.s52e01.20220214.Labi_Siffre_This_Is_My_Song_.xml-   <series>2022</series>
imagine.s52e01.20220214.Labi_Siffre_This_Is_My_Song_.xml:   <seriesnum>52</seriesnum>
--
imagine.s52e01.20220221.Wayne_McGregor_Dancing_on_the_Edge_.xml-    <series>2022</series>
imagine.s52e01.20220221.Wayne_McGregor_Dancing_on_the_Edge_.xml:    <seriesnum>52</seriesnum>
--
imagine.s52e01.20220425.Miriam_Margolyes_Up_for_Grabs_.xml- <series>2022</series>
imagine.s52e01.20220425.Miriam_Margolyes_Up_for_Grabs_.xml: <seriesnum>52</seriesnum>
--
imagine.s52e01.20221031.Malorie_Blackman_What_If_.xml-  <series>2022</series>
imagine.s52e01.20221031.Malorie_Blackman_What_If_.xml:  <seriesnum>52</seriesnum>
--
imagine.s52e01.20221107.Sonia_Boyce_Finding_Her_Voice_.xml- <series>2022</series>
imagine.s52e01.20221107.Sonia_Boyce_Finding_Her_Voice_.xml: <seriesnum>52</seriesnum>
--
imagine.s52e01.20221114.Douglas_Stuart_Love_Hope_and_Grit_.xml- <series>2022</series>
imagine.s52e01.20221114.Douglas_Stuart_Love_Hope_and_Grit_.xml: <seriesnum>52</seriesnum>
--
imagine.s53e01.20230320.Stephen_Frears_Director_for_Hire_.xml-  <series>2023</series>
imagine.s53e01.20230320.Stephen_Frears_Director_for_Hire_.xml:  <seriesnum>53</seriesnum>
--
imagine.s53e01.20231211.The_Factory_Made_in_Manchester_.xml-    <series>2023</series>
imagine.s53e01.20231211.The_Factory_Made_in_Manchester_.xml:    <seriesnum>53</seriesnum>
--
imagine.s53e01.20231227.French_Saunders_Pointed_Bitchy_Bitter_.xml- <series>2023</series>
imagine.s53e01.20231227.French_Saunders_Pointed_Bitchy_Bitter_.xml: <seriesnum>53</seriesnum>